Control system and method for a rotating electromagnetic machine

ABSTRACT

A system and method for controlling a rotating electromagnetic machine. The rotating machine, such as a permanent magnet motor or switched reluctance motor, includes a stator having a plurality of phase windings and a rotor that rotates relative to the stator. A drive is connected to the phase windings for energizing the windings. The control system includes an estimator connectable to the machine for receiving signals representing the phase winding voltage and rotor position. The estimator outputs parameter estimations for an electrical model of the machine based on the received voltage and rotor position. A torque model receives the parameter estimations from the estimator to estimate torque for associated rotor position-phase current combinations of the machine. A controller outputs a control signal to the drive in response to a torque demand and rotor position signals and the torque model. In certain embodiments, a solver uses the torque model to generate energization current profiles according to desired machine behavior, such as smooth torque and/or minimal sensitivity to errors in rotor position measurement.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates generally to the control of electromagneticrotating machines such as permanent magnet, switched reluctance andhybrid machines thereof, and more particularly, to adaptive, smoothtorque control of such machines.

2. Description of Related Art

Many electromagnetic machines in general, and electric motors employingpermanent magnets in particular, exhibit torque irregularities as therotor rotates with respect to the stator, the coils of which aretypically energized with a sinusoidal waveform. Such irregularities arereferred to as “torque ripple.” These torque irregularities may becaused by the physical construction of a given machine. For example,they may result from the use of bearings to support the rotor. Inaddition, because of the electromagnetic characteristics of machinesthat employ magnets, the rotor tends to prefer certain angular positionswith respect to the stator. Torque irregularities resulting from theelectromagnetic characteristics of an electromagnetic machine arecommonly known as “cogging” irregularities and the resultant non-uniformrotation of the rotor or non-uniform torque output is known as“cogging.” Cogging is either current independent or current dependent.The first component is noted when the machine is spun unenergized. Thesecond component is present when current flows—the cogging grows as themagnitude of the stator currents increases.

In rotating electromagnetic machines employing permanent magnets,cogging most often results from the physical construction of themachine. Irregularities due to the magnets can result, for example, fromthe magnets being incorrectly placed upon or in the rotor, or from someirregularity about how the magnets are energized. Moreover, theutilization of rotors having discrete north and south outer polesresults in a circumferential distribution of magnetic flux about therotor circumference that is not smooth, but choppy. Additionally, thestators commonly used with such machines are formed in such a way thatthe magnetic fluxes generated by the stator windings provide a fluxdistribution about the stator circumference that is not smooth. Thecombination of such rotors and stators, and the accompanying non-smoothflux distributions, produces undesired irregularities in the torqueoutput of such machines. Rotor output non-uniformities may also beproduced by back emf harmonics produced in certain machines.

Obtaining smooth torque is further complicated by other factors. Forinstance, manufacturing variances between motors makes it difficult, ifnot impossible, to apply a common solution to a group of motors. Suchmanufacturing variance includes the placing or misplacing of magnetsupon the rotor (if surface mounted), variance introduced by themagnetizing process itself and irregularities in the stator coilwindings. Other causes of variance include instances when the magnetsare damaged or chipped. Further, variations exist even within individualmotors. For example, variations typically exist between a motor's phasesand over the motor's full mechanical cycle. Moreover, motor behaviorchanges over time as the motor ages.

The phase windings of certain types of electromagnetic machines areenergized at least in part as a function of the instantaneous rotorposition. Accordingly, such machines often use a rotor position sensorthat provides an output indicating rotor position relative to thestator. A controller uses this information to produce control signalsthat are used to energize and de-energize the phase windings. Errors inthe measurement of the angular position of the rotor also contribute totorque ripple.

For many motor applications a slight non-uniformity in the rotation ofthe rotor caused by torque irregularities is of little or noconsequence. For example, in large motors driving large loads, slightvariations in the output torque will not significantly affect the rotorspeed and any slight variations in rotor speed will not significantlyaffect the system being driven by the machine. This assumes that thetorque variation as the machine turns is small compared to the load. Inother applications, where the rotation of the rotor or the torque outputof the motor must be precisely controlled or uniform, suchnon-uniformity is not acceptable. For example, in servomotors used inelectric power steering systems and is in disk drives, the rotationaloutput of the rotor or the torque output of the motor must be smooth andwithout significant variation.

Prior art approaches to reducing the undesirable consequences of torqueirregularities in electromagnetic machines have focused on relativelycomplex rotor or stator constructions designed to eliminate the physicalcharacteristics of the machines that would otherwise give rise to theirregularities. While the prior art machine construction approaches canresult in reduction of torque irregularities, the approaches require thedesign and construction of complex rotor and stator components, suchcomplex components are typically difficult to design, difficult tomanufacturer, and much more costly to produce than are conventionallyconstructed components. Moreover, many of the physical changes requiredby such prior art solutions result in a significant reduction in theefficiency or other performance parameters of the resulting machinesover that expected of comparable conventional machines. Thus, many ofthe prior art attempts to reduce torque irregularities do so at the costof machine performance.

Attempts to reduce torque ripple that focus on motor control schemes,rather than motor construction, have also been undertaken. For example,various learning or iterative schemes, based on either experimentalprocedures or well known physical relationships concerning motorvoltages, currents and angular positions have been attempted. Theseattempted solutions often make assumptions concerning the behavior ofthe motor, such as motor flux being described by a linear relationship,or considering the effect of mutual flux insignificant. Still further,prior solutions to torque ripple typically ignore the effect of motorsensitivity to inaccuracies in angular measurement. To increase accuracyin position measurement, the use of sophisticated position sensors hasbeen attempted, but this increases the machine's complexity and cost.

Thus, a need exists for a control system that addresses the shortcomingsof the prior art.

SUMMARY OF THE INVENTION

In one aspect of the present invention, a system for controlling arotating electromagnetic machine is presented. The rotating machine,such as a permanent magnet motor or switched reluctance motor, or somehybrid of the two, includes a stator having a plurality of phasewindings and a rotor that rotates relative to the stator. A drive isconnected to the phase windings for energizing the windings. The controlsystem includes an estimator connectable to the machine for receivingsignals representing the phase winding voltage, current and rotorposition. The estimator outputs parameter estimations for an electricalmodel of the machine based on the received voltage, current and rotorposition. The electrical model is a mathematical model that describeselectrical behavior of the machine as seen at the motor terminals.

A torque model receives the parameter estimations from the estimator.The torque model is developed via a mathematical transform of theelectrical model, and describes torque characteristics of the machine.Using the parameters received from the estimator, the torque modelestimates torque for associated rotor position-phase currentcombinations. A controller has input terminals for receiving a torquedemand signal and the rotor position signal. The controller outputs acontrol signal to the drive in response to the torque demand and rotorposition signals and the torque model. In certain embodiments, a solveruses the torque model to generate energization current profilesaccording to desired machine behavior, such as smooth torque and/orminimal sensitivity to errors in rotor position measurement. It is to benoted in particular that the solver can be so defined that the solutionpossesses particular properties. It may be desired to deal with only themost significant components of cogging or torque ripple, a result ofmotor drive cost considerations. Such a solution can be achieved.

Some parameters of the torque model are unobservable via informationimmediately available from the machine terminals. For example, inmachines employing permanent magnets, it is not mathematically obvioushow changes in the machine current and voltage, as the rotor spins,indicate or measure the interaction of the machine's magnets withthemselves. In accordance with further aspects of the present invention,a method of determining a non-load dependent cogging torque is provided.The rotor is spun unloaded at a predetermined angular velocity and themotor terminal voltage and current are measured. Rotor positionsassociated with the voltage and current measurements are determined, anda first mathematical model is developed based on the measured voltageand rotor position to describe electrical behavior of the machine. Thefirst mathematical model is mathematically transformed to develop asecond mathematical model to describe torque characteristics of themachine. The windings are then energized such that the rotor holds apredetermined position against the cogging torque, and the coggingtorque is calculated for the predetermined position via the secondmathematical model.

BRIEF DESCRIPTION OF THE DRAWINGS

Other objects and advantages of the invention will become apparent uponreading the following detailed description and upon reference to thedrawings in which:

FIG. 1 is a block diagram of an exemplary rotating electromagneticmachine system in accordance with embodiments of the present invention;

FIG. 2 is a block diagram illustrating an electromagnetic machinecontrol system in accordance with embodiments of the present invention;

FIG. 3 is a chart illustrating an exemplary integration path forevaluating an integral giving coenergy used in embodiments of theinvention;

FIG. 4 illustrates current profiles generated for a smooth torquesolution in accordance with the present invention; and

FIG. 5 illustrates torque ripple for a machine controlled in accordancewith aspects of the present invention.

While the invention is susceptible to various modifications andalternative forms, specific embodiments thereof have been shown by wayof example in the drawings and are herein described in detail. It shouldbe understood, however, that the description herein of specificembodiments is not intended to limit the invention to the particularforms disclosed, but on the contrary, the intention is to cover allmodifications, equivalents, and alternatives falling within the spiritand scope of the invention as defined by the appended claims.

DETAILED DESCRIPTION OF THE INVENTION

Illustrative embodiments of the invention are described below. In theinterest of clarity, not all features of an actual implementation aredescribed in this specification. It will of course be appreciated thatin the development of any such actual embodiment, numerousimplementation-specific decisions must be made to achieve thedevelopers' specific goals, such as compliance with system-related andbusiness-related constraints, which will vary from one implementation toanother. Moreover, it will be appreciated that such a development effortmight be complex and time-consuming, but would nevertheless be a routineundertaking for those of ordinary skill in the art having the benefit ofthis disclosure.

Turing to the drawings and, in particular, to FIG. 1, a system 10constructed according to certain teachings of this disclosure isillustrated. Among other things, the illustrated system 10 activelycontrols the electric power supplied to an electromagnetic machine suchthat the negative consequences of torque irregularities that wouldotherwise be produced by the machine are reduced or eliminated.

The system 10 includes an electromagnetic machine 12 and a drive 14 thatprovides electric power to the electromagnetic machine 12. The machine12 shown in FIG. 1 may comprise, for example, a permanent magnet motor,a switched reluctance motor, or a hybrid motor (permanent magnet andswitched reluctance combination). The machine 12 is of conventionalconstruction that includes a rotating component (a rotor 12 a) and astationary component (a stator 12 b). Wound about the stator are anumber of energizable phase windings 12 c which may be energized throughthe application of electric power to motor terminals 15, 16, 17.

The drive 14 is coupled to provide electric power to terminals 15, 16and 17 of the machine 12. The drive 14 receives control inputs from acontrol system 13, which is coupled to receive feedback from the machine12 in terms of rotor position information 18 and energization feedback19. Other feedback information may be provided to the controller 13.While the drive 14 is illustrated in exemplary form as providing threepower terminals to the machine 12, it should be understood that more orfewer power terminals may be provided to accommodate motors or machineswith greater than three phases, less than three phases or if varioustypes of inverters (e.g., with neutral connections) are used.

The energization feedback 19 provides an indication of the operationalcharacteristics of the machine 12 and may, for example, include feedbackconcerning the currents flowing in the stator windings and/or thevoltages at the terminals 15, 16 and 17. The position and energizationparameters may be detected through conventional detectors such asstandard rotor position detectors and standard current/voltage sensors.Alternative embodiments are envisioned in which the rotor position andfeedback parameters are not detected directly but are calculated orestimated through known techniques. For example, embodiments areenvisioned where only the terminal voltages are known or sensed alongwith the currents flowing through the stator windings of the machine 12and the sensed current and voltage values are used to derive rotorposition information.

The control system 13 also receives input command signals 11 thatcorrespond to a desired output parameter of machine 12 such as rotorspeed, output torque, etc. As described in more detail below, the drive14 controls the application of electric power to machine 12 in responseto the control system 13 in such a manner that the difference betweenthe input command signal 11 and the corresponding output of the machine12 is minimized. In certain embodiments, the control system 13 alsoactively controls application of power to the machine 12 as a functionof rotor position in such a manner to achieve a desired behavior of themachine 12 meeting one or more criteria in categories including, forexample, torque ripple, cogging torque, angular sensitivity, harmoniccontent, etc. The use of the control system 13 to actively achievedesired machine behavior, as opposed to attempting to achieve suchbehavior through complex rotor or stator constructions, results in abetter performing system in that, for example, conventional, low costmachines and machine construction techniques may be used.

An electromagnetic machine system 100 in accordance with an exemplaryembodiment of the present invention is shown in FIG. 2. The machinesystem 100 includes a control system 13, which may be implemented by anappropriately programmed digital controller, such as a digital signalprocessor (DSP), microcontroller or microprocessor.

The control system 13 includes an input terminal 11 that receives, forexample, a signal representing the torque demanded of the motor 12.Torque is a function of current and angle; hence, for any particularrotor angle there is a set of appropriate currents that will produce thedesired torque. Based on the rotor angle and the required torque,appropriate current values are sent to the drive 14, which in turnprovides the necessary voltage to the motor 12 to meet the currentdemand.

Rotor position feedback 18 and energization feedback 19, such as themotor terminal voltage and current, are provided to an estimator 30. Inaccordance with mathematical “good practice,” the voltage and currentvalues may be normalized—the measured values are divided by the maximumexpected value. The estimator 30 calculates motor parameters such asangular speed and the time derivatives of the phase currents. Thesevalues are used to derive and update a first mathematical model thatdescribes the electrical behavior of the motor 12. The structure of theelectrical model is such that it can accurately represent electricalmachine characteristics such as resistance, back electromagnetic force(“BEMF”), self and mutual inductance, cogging, etc., depending on thetype of machine 12 employed.

The parameters calculated by the estimator 30 are passed to a torquemodel 32 of the motor 12. The torque model 32 is developed bymathematically transforming the electrical model in an appropriatemanner, dictated by the electromagnetic physics of the motor 12, into asecond model that describes the torque characteristics of the machine12. Since the electrical model coefficients flow naturally into thetorque model 32, by constructing an accurate electrical model of themotor 12, the torque characteristics of the motor 12 are also known. Forexample, the torque model may describe the torque produced for anycombination of phase current and rotor position in the normal operatingenvelope of the machine. Thus, using the values calculated by theestimator 30, an estimate of motor torque can be calculated for anycurrent-angle combination.

The torque model 32 is interrogated by a solver 34, which calculates therequired currents, or solution curves, according to some desired motorbehavior and the known behavior of the motor 12 (as regards current andangle). Thus, the controller 36 provides the appropriate current for agiven rotor angular position to achieve the desired output torque 40 orother output parameter, and further, to achieve the output parameter inaccordance with the desired machine behavior. For example, the desiredmachine behavior may include the operating characteristics of the motor12 meeting one or more criteria in categories including cogging torque,torque ripple, angular sensitivity of the solution to angular error andharmonic content of the solution curves. In the particular system 100shown in FIG. 2, the solver 34 output is stored explicitly as a lookuptable accessible by the controller 36. The torque demand 11 and rotorangle is applied to the lookup table to determine the appropriate phasecurrent value to be applied to the phase windings via the driver 14. Inother embodiments, the output of the solver 34 is in an analytic form,derived by fitting a function to the calculated numerical values.

Since the electrical model used by the estimator 30 is algebraic innature, the estimator 30 can be allowed to run for some time periodoperating upon not necessarily sequential data before a new set ofparameter estimates are released in to the torque model 32. At thispoint the solver 34 can then recalculate the necessary lookup tables 36.Once fully calculated, the new lookup tables can then replace thosetables currently in use. Many of these operations can be backgroundtasked; that is, they can occur as and when computational resources areavailable. This is one of the advantages of an algebraic motor model—itis not intricately wrapped up in the time variable.

Torque can be estimated via coenergy or field energy, though calculatingtorque via coenergy results in simpler expressions. Thus, the estimateof output torque 40 can be calculated using only feedback available fromthe machine terminals—such as the terminal voltage and current and therotor position—to estimate the parameters of resistance and fluxlinkage. The following disclosure is generally provided in terms of athree phase hybrid motor, though the model form can be generalized intodifferent types of rotating machines having any number of phases by oneskilled in the art having the benefit of this disclosure.

It is common in many applications to utilize what is known as a balancedthree phase feed. In such systems, when a three-phase motor is used, thesum of the three phase currents will equal zero. Hence, the αβ-Frame ofReference (FoR) can be used. If balanced feed is not used, it isnecessary to use the abc-FoR. The αβ-FoR is considered first.

The electrical model may have the form of a product of polynomialexpressions in current and angle. Typically, those for angle willinvolve trigonometric functions. The current polynomials may also beorthogonal and may be one of any number of suitable polynomial types.For more complex machines, an orthogonal model form may be appropriate.With the first model structure disclosed herein, it is assumed that fluxlinkage models are expressions that are products of polynomial termsinvolving phase currents and trigonometric polynomials of mechanicalangle. Models using orthogonal functions are discussed further later inthis specification. Generally, the following nomenclature is used inthis disclosure:

φ is the phase index, ranges over defined set of numbers {1,2,3, . . . }or equivalent letters {a,b,c, . . . }

a,b,c are phase names, equivalent to 1,2,3 when numerically referenced

α,β,0 are αβ Frame of Reference (FoR) labels

λ_(φ) is the phase φ flux linkage

p,P . . . q,Q . . . r,R . . . n,N are summation indexes and maximumindex values

sin( ), cos( ) are trigonometric functions

g_(φpqrn), h_(φpqrn) are model parameters

i_(α),i_(β),i_(C) are variables representing αβ frame of referencecurrents

I_(α),I_(β) are maximum values of αβ-frame of reference currentsencountered in the coenergy integral

i_(a),i_(b),i_(c) are variables representing abc-frame of referencecurrents

I_(a),I_(b),I_(c) are maximum values of abc-FoR currents encountered inthe coenergy integral

i_(f) is the current flow associated with the fictious rotor circuitmodelling the presence of a magnet

v_(φ) is the phase φ voltage

R_(φα),R_(φβ),R_(φαβ) are resistance values associated with phase φ$\frac{\quad}{y}x$

is the differential of x with respect to y

θ is the rotor angle

ω is the rotor angular velocity

t is time t

ω_(c) is coenergy

dx is x infintesimal

∫ f(x) dx is the integral of f(x) with respect to x$\frac{\partial\quad}{\partial x}{f\left( {x,y,\ldots} \right)}$

is the partial derivative of function f(.) with respect to x

D₁, . . . ,D₆ are components of the coenergy integral, along definedpath

T is motor or true torque

o(x^(n)) is the remainder associated with n'th order and higher terms ofthe Jacobian matrix

J_(ij) is the ij'th entry of the Jacobian

F_(i)(x₁, . . . ,x_(M)) is the i'th function of variables x₁, . . .,x_(M) δx is delta x

δx_(new) is the new value of delta x, or change in x

x_(new), x_(old) are new and old values of x calculated during Newtoniterative process

T_(tv) is estimated torque, directly from the terminal variables

T_(cog) is torque which cannot be calculated directly using terminalvariables

S_(a) is solution sensitivity $I = \begin{pmatrix}{i_{\alpha}\left( {\theta (1)} \right)} \\{i_{\alpha}\left( {\theta (2)} \right)} \\\cdots \\{i_{\alpha}\left( {\theta (N)} \right)} \\{i_{\beta}\left( {\theta (1)} \right)} \\{i_{\beta}\left( {\theta (2)} \right)} \\\cdots \\{i_{\beta}\left( {\theta (N)} \right)}\end{pmatrix}$

is the vector of αβ frame of reference current values across the set ofdiscrete angle values

I(n+1),I(n) are the (n+1)'th and n'th iterated current vectors

ΔI(n) is the calculated change in current vector at n'th interval

φ_(Tk)=(0 . . . 0 T(θ(k),i_(α)(k),i_(β)(k)) 0 . . . 0) is the k'thtorque vector

φ_(Sk)=(0 . . . 0 S(θ(k),i_(α)(k),i_(β)(k)) 0 . . . 0) is the k'thsensitivity vector ${A = \begin{pmatrix}\varphi_{T1} \\\cdots \\\varphi_{TN}\end{pmatrix}}\quad$

are stacked torque vectors $B = \begin{pmatrix}\varphi_{S1} \\\cdots \\\varphi_{SN}\end{pmatrix}$

are stacked current vectors

Assuming that the model structure for each machine phase (φ) isidentical, the general form of the flux model using the αβ-FoR is:$\begin{matrix}{\lambda_{\varphi} = {\sum\limits_{p = 0}^{P}\quad {i_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}\quad {i_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {i_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{g_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}}} & (1)\end{matrix}$

Such a model allows for a non-linear relationship between phase currentand flux as well as mutual effects between any two or more phases. Asnoted above, for the purposes of the present disclosure it is assumedthat model structure is invariant with respect to phase, although thisneed not be so. Contiguous powers of polynomial current and angleharmonic need not be used, as is the case in Equation (1). For example,consider the following: $\begin{matrix}{\lambda_{\varphi} = {\sum\limits_{p = p_{1}}^{p_{S}}\quad {i_{\alpha}^{p} \cdot {\sum\limits_{q = q_{1}}^{q_{T}}\quad {i_{\beta}^{q} \cdot {\sum\limits_{r = r_{1}}^{r_{U}}\quad {i_{f}^{r} \cdot {\sum\limits_{n = n_{1}}^{n_{V}}\quad \left( {{g_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}}} & (2)\end{matrix}$

where the indexing sets:

p=(p₁,p₂, . . . ,p_(S).) r=(r₁,r₂, . . . ,r_(U).)

q=(q₁,p₂, . . . ,q_(T).) n=(n₁,n₂, . . . ,n_(V).)

need not contain contiguous integers. In fact, most practicalapplications will have this form.

Relatively simple models of the form presented in Equation (2) that aresufficiently accurate can be obtained. Model structure can be allowed tovary between phases if so desired. This variation upon defining modelstructure has a significant impact upon the computational complexity ofthe associated algorithms. Some model components will be present as aresult of manufacturing variance and would not be suggested by atheoretical consideration of the motor design. Further, model complexitycan vary greatly between motors of different design. For example, apermanent magnet motor design with the express intent of reducingcogging, typically through the use of skew, may only require a verysimple model to accurately predict torque. It is generally desirable toavoid models that are over or under determined.

The abc-FoR currents can be transformed into αβ-FoR currents using thefollowing transform: $\begin{matrix}{\begin{pmatrix}i_{\alpha} \\i_{\beta} \\i_{0}\end{pmatrix} = {\begin{pmatrix}1 & 0 & 1 \\\frac{- 1}{2} & {\frac{- 1}{2} \cdot \sqrt{3}} & 1 \\\frac{- 1}{2} & {\frac{1}{2} \cdot \sqrt{3}} & 1\end{pmatrix} \cdot \begin{pmatrix}i_{\alpha} \\i_{b} \\i_{c}\end{pmatrix}}} & (3)\end{matrix}$

Under the balanced feed assumption, the third phase current is zero. Itis known that phase voltage (v_(φ)) is defined by $\begin{matrix}{v_{\varphi} = {{i_{\varphi} \cdot R_{\varphi}} + {\frac{\quad}{t}\lambda_{\varphi}}}} & (4)\end{matrix}$

where R_(φ) is the phase resistance. Thus, using the αβ-FoR:$\begin{matrix}{v_{\varphi} = {R_{\varphi} + {i_{\alpha} \cdot R_{\varphi\alpha}} + {i_{\beta} \cdot R_{\varphi\beta}} + {i_{\alpha} \cdot i_{\beta} \cdot R_{{\varphi\alpha}\quad \beta}} + {\frac{\quad}{t}\lambda_{\varphi}}}} & (5)\end{matrix}$

It should be noted that there are more resistance terms in Equation (5)then is necessary from the perspective of how electric circuits operate.Such additional terms allow for the presence of test data offsets andthe like to be directly compensated for; otherwise the estimator willset redundant terms to zero.

It is also known that angular velocity (ω) is defined by:$\begin{matrix}{\omega = {\frac{\quad}{t}\theta}} & (6)\end{matrix}$

From Equations (1) and (4): $\begin{matrix}{{{\frac{\quad}{t}\lambda_{\varphi}} = {{\sum\limits_{p = 1}^{P}\quad {\left( {{{pi}_{\alpha}^{p - 1} \cdot \frac{\quad}{t}}i_{\alpha}} \right) \cdot {\sum\limits_{q = 0}^{Q}{i_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}{\left( {{g_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \cdots}}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{\alpha}^{p} \cdot {\sum\limits_{q = 1}^{Q}{\left( {{{qi}_{\beta}^{q - 1} \cdot \frac{\quad}{t}}i_{\beta}} \right) \cdot {\sum\limits_{r = 0}^{R}{i_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}{\left( {{g_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \cdots}}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{\beta}^{q} \cdot {\sum\limits_{r = 1}^{R}{\left( {{{ri}_{f}^{r - 1} \cdot \frac{\quad}{t}}i_{f}} \right) \cdot {\sum\limits_{n = 0}^{N}{\left( {{g_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \cdots}}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{f}^{r} \cdot {\sum\limits_{n = 1}^{N}{\omega \cdot n \cdot \left( {{g_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}}\quad} & (7)\end{matrix}$

The imaginary rotor current (i_(f)) is nominally constant and its timederivative is zero. Hence, from Equations (5) and (7): $\begin{matrix}{\quad {v_{\varphi} = {R_{\varphi} + {i_{\alpha} \cdot R_{\varphi\alpha}} + {i_{\beta} \cdot R_{\varphi\beta}} + {i_{\alpha} \cdot i_{\beta} \cdot R_{{\varphi\alpha\beta}\quad \cdots}} + \quad {\sum\limits_{p = 1}^{P}\quad {\left( {{{pi}_{\alpha}^{p - 1} \cdot \frac{\quad}{t}}i_{\alpha}} \right) \cdot {\sum\limits_{q = 0}^{Q}{i_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}{\left( {{g_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \cdots}}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{\alpha}^{p} \cdot {\sum\limits_{q = 1}^{Q}{\left( {{{qi}_{\beta}^{q - 1} \cdot \frac{\quad}{t}}i_{\beta}} \right) \cdot {\sum\limits_{r = 0}^{R}{i_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}{\left( {{g_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \cdots}}}}}}}} + \quad \quad {\sum\limits_{p = 0}^{P}{i_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{f}^{r} \cdot {\sum\limits_{n = 1}^{N}{\omega \cdot n \cdot \left( {{g_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}}\quad} & (8)\end{matrix}$

In embodiments employing a switched reluctance machine, there is noimaginary rotor current state as there are no rotor magnets with whichthis state is associated, hence: $\begin{matrix}{i_{f} \equiv {0\quad {and}\quad \frac{\quad}{t}i_{f}} \equiv 0} & (9)\end{matrix}$

Therefore, in the case of a switched reluctance machine, indexingvariable (r) associated with the imaginary rotor phase may be removedfrom Equation (8): $\begin{matrix}{{v_{\varphi} = {R_{\varphi} + {i_{\alpha} \cdot R_{\varphi\alpha}} + {i_{\beta} \cdot R_{\varphi\beta}} + {i_{\alpha} \cdot i_{\beta} \cdot R_{{\varphi\alpha\beta}\quad \cdots}} + {\sum\limits_{p = 1}^{P}\quad {\left( {{{pi}_{\alpha}^{p - 1} \cdot \frac{\quad}{t}}i_{\alpha}} \right) \cdot {\sum\limits_{q = 0}^{Q}{i_{\beta}^{q} \cdot {\sum\limits_{n = 0}^{N}{\left( {{g_{\varphi \quad {pqn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \cdots}}}}}} + \quad \quad {\sum\limits_{p = 0}^{P}{i_{\alpha}^{p} \cdot {\sum\limits_{q = 1}^{Q}{\left( {{{qi}_{\beta}^{q - 1} \cdot \frac{\quad}{t}}i_{\beta}} \right) \cdot {\sum\limits_{n = 0}^{N}{\left( {{g_{\varphi \quad {pqn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \cdots}}}}}} + \quad {\sum\limits_{p = 0}^{P}{i_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{\omega \cdot n \cdot \left( {{g_{\varphi \quad {pqn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{\varphi \quad {pqn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}\quad} & (10)\end{matrix}$

It would be a routine undertaking for one skilled in the art having thebenefit of this disclosure to apply a similar process of simplificationto the specific case of an SR motor, for example.

There are several techniques available for calculating the coefficientsin the model given test data. As noted above, the electrical model isalgebraic, which allows the use of any of a number of parameterestimation techniques, such as least squares methods or grammian matrixmethods. Least squares-based parameter estimators find the modelcoefficients that minimize the square of the difference between theobserved data and the output from the model. Recursive least squareparameter estimators are used in particular embodiments of theinvention. The recursive least squares parameter estimators are mostsuitable for actual production systems. They operate in such a mannerthat they can produce an improved estimate with each new sample of data.That is, they are not restricted in their operation to complete sets oftest data.

A further refinement of the recursive least squares parameter estimationtechnique involves the use of a “forgetting factor,” which operates inthe following manner. As more and more data is captured, the effect thatthe old data has upon the calculation of the new data is reduced. Inthis way, only the most recent data will have a significant effect inthe parameter estimation process. This forgetting factor operates overany appropriate time interval—for example, minutes, hours ordays—depending on design considerations. Many of the variablesconsidered do not vary significantly over time. Some, however, such asphase resistance, vary over time and with respect to other factors suchas the machine temperature. This added refinement allows the controlsystem to be tuned to any particular machine, and also allows adaptationto changes in that machine. This typically occurs as the machine ages.

Various data collection schemes may be used for parameter estimation.For example, one data collection technique requires constant phasecurrent, which usually only occurs in controlled data collectionsituations. Another involves varying current, typical of practicalapplications.

In the constant phase current data collection scheme,${\frac{\quad}{t}i_{\alpha}} = {{\frac{\quad}{t}i_{\beta}} = 0}$

Under these conditions, Equation (8) reduces to: $\begin{matrix}{{v_{\varphi} = {R_{\varphi} + {i_{\alpha} \cdot R_{\varphi\alpha}} + {i_{\beta} \cdot R_{\varphi\beta}} + {i_{\alpha} \cdot i_{\beta} \cdot R_{{\varphi\alpha\beta}\quad \cdots}} + {\sum\limits_{p = 0}^{P}{i_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{f}^{r} \cdot {\sum\limits_{n = 1}^{N}{\omega \cdot n \cdot \left( {{g_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}}\quad} & (11)\end{matrix}$

For notational convenience and to reflect the unobservability of thei_(f) term, the following identities are defined as a result ofconsidering the last two Σ terms in Equation (11): $\begin{matrix}{{{\sum\limits_{r = 0}^{R}{i_{f}^{r} \cdot g_{\varphi \quad {pqrn}}}} = {G_{\varphi \quad {pqn}}\quad {for}\quad {all}\quad \varphi}},{p\quad {and}\quad q}} & (12) \\{{{\sum\limits_{r = 0}^{R}{i_{f}^{r} \cdot h_{\varphi \quad {pqrn}}}} = {H_{\varphi \quad {pqn}}\quad {for}\quad {all}\quad \varphi}},{p\quad {and}\quad q}} & (13)\end{matrix}$

To retain consistency with Equation (10) for the case in which phasecurrent varies, the harmonic terms (n) in Equation (11) are notcollected into the H and G terms defined by Equations (12) and (13).Substituting Equations (12) and (13) into Equation (11) results in:$\begin{matrix}{{v_{\varphi} = {R_{\varphi} + {i_{\alpha} \cdot R_{\varphi\alpha}} + {i_{\beta} \cdot R_{\varphi\beta}} + {i_{\alpha} \cdot i_{\beta} \cdot R_{{\varphi\alpha\beta}\quad \cdots}} + {\sum\limits_{p = 0}^{P}{i_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{\omega \cdot n \cdot \left( {{G_{\varphi \quad {pqn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{\varphi \quad {pqn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}\quad} & (14)\end{matrix}$

Division throughout Equation (14) by ω yields: $\begin{matrix}{{\frac{v_{\varphi}}{\omega} = {{\frac{1}{\omega} \cdot R_{\varphi}} + {\frac{i_{\alpha}}{\omega} \cdot R_{\varphi\alpha}} + {\frac{i_{\beta}}{\omega} \cdot R_{\varphi\beta}} + {\frac{i_{\alpha} \cdot i_{\beta}}{\omega} \cdot R_{{\varphi\alpha\beta}\quad \cdots}} + {\sum\limits_{p = 0}^{P}{i_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{\varphi \quad {pqn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{\varphi \quad {pqn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}\quad} & (15)\end{matrix}$

Using Equation (14), the motor parameters can be estimated in the caseof constant phase current.

As noted above, practical motor applications involve varying current.Equation (8) describes how the flux model coefficients, first presentedin Equation (1), propagate through three separate paths when calculatingphase voltage in the variable current case. The first path, aspreviously encountered, passes through those expressions explicitlyinvolving ω. The other two paths involve expressions with timederivatives of the currents. As before, ω is divided throughout andsubstitutions as indicated in Equations (12) and (13) are made,yielding: $\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{\frac{v_{\varphi}}{\omega}\quad = {{\frac{1}{\omega} \cdot R_{\varphi}} + {\frac{i_{\alpha}}{\omega} \cdot R_{\varphi \quad \alpha}} + {\frac{i_{\beta}}{\omega} \cdot R_{\varphi \quad \beta}} + {{\frac{i_{\alpha} \cdot i_{\beta}}{\omega} \cdot R_{\varphi \quad \alpha \quad \beta}}\ldots} +}} \\{\quad {{\left\lbrack {\frac{1}{\omega} \cdot \left( {\frac{\quad}{t}i_{\alpha}} \right)} \right\rbrack \cdot {\sum\limits_{p = 1}^{P}\quad {{pi}_{\alpha}^{p - 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {i_{\beta}^{q} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{G_{\varphi \quad {pqn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {H_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\ldots}}}}}}} +}} \\{\quad {{\left\lbrack {\frac{1}{\omega} \cdot \left( {\frac{\quad}{t}i_{\beta}} \right)} \right\rbrack \cdot {\sum\limits_{p = 0}^{P}\quad {i_{\alpha}^{p} \cdot {\sum\limits_{q = 1}^{Q}\quad {{qi}_{\beta}^{q - 1} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{G_{\varphi \quad {pqn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {H_{\varphi \quad {pqn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\ldots}}}}}}} +}} \\{\quad {\sum\limits_{p = 0}^{P}\quad {i_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}\quad {i_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \quad \left( {{G_{\varphi \quad {pqn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{\varphi \quad {pqn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}\end{matrix} \\{{Note}\quad {{that}:}}\end{matrix} \\{{\frac{1}{\omega} \cdot \left( {\frac{\quad}{t}i} \right)} = {{{\frac{1}{\left( {\frac{\quad}{t}\theta} \right)} \cdot \frac{\quad}{t}}i} = {{\left( {\frac{\quad}{\theta}t} \right) \cdot \left( {\frac{\quad}{t}i} \right)} = {\frac{\quad}{\theta}i}}}}\end{matrix} & (16)\end{matrix}$

This allows a simple re-working of Equation (16) as follows:$\begin{matrix}{\frac{v_{\varphi}}{\omega}\quad = {{\frac{1}{\omega} \cdot R_{\varphi}} + {\frac{i_{\alpha}}{\omega} \cdot R_{\varphi \quad \alpha}} + {\frac{i_{\beta}}{\omega} \cdot R_{\varphi \quad \beta}} + {{\frac{i_{\alpha} \cdot i_{\beta}}{\omega} \cdot R_{\varphi \quad \alpha \quad \beta}}\ldots} +}} \\{\quad {{\left( {\frac{\quad}{\theta}i_{\alpha}} \right) \cdot {\sum\limits_{p = 1}^{P}\quad {{pi}_{\alpha}^{p - 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {i_{\beta}^{q} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{G_{\varphi \quad {pqn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {H_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\ldots}}}}}}} +}} \\{\quad {{\left( {\frac{\quad}{\theta}i_{\beta}} \right) \cdot {\sum\limits_{p = 0}^{P}\quad {i_{\alpha}^{p} \cdot {\sum\limits_{q = 1}^{Q}\quad {{qi}_{\beta}^{q - 1} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{G_{\varphi \quad {pqn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {H_{\varphi \quad {pqn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\ldots}}}}}}} +}} \\{\quad {\sum\limits_{p = 0}^{P}\quad {i_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}\quad {i_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \quad \left( {{G_{\varphi \quad {pqn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{\varphi \quad {pqn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}\end{matrix}$

The advantage of this re-formulation of Equation (16) is that twopossible sources of impulsive noise in the equation, and the problemsthey can cause with a parameter estimator, have been removed. If thedriving current forms are assumed known, then their derivatives withrespect to angle can be directly calculated. If the current forms aredefined in an analytic form, such as from a BEMF model, then a closedexpression exists for the derivative else a numerical estimate can beobtained. In particular, if at the (k−1) and k intervals the angle anddesired currents are θ(k−1), θ(k), i_(α)(k−1) and i_(α)(k) then:${\frac{\quad}{\theta}i_{\alpha}} = \frac{{i_{\alpha}(k)} - {i_{\alpha}\left( {k - 1} \right)}}{{\theta (k)} - {\theta \left( {k - 1} \right)}}$

This approach implicitly assumes the system exhibits good currentfollowing properties.

Since resistance and flux linkage can be estimated via the electricalmodel, a torque model relating current and angle to torque generated canbe derived through a series of standard operations. Very generally,these operations include integrating the flux linkage from zero currentto the present value of current—coenergy, and differentiating thisexpression with respect to shaft angle. The result of these operationsis an expression for torque.

Coenergy (ω_(c)) is defined by: $\begin{matrix}{\omega_{c} = {\int{\sum\limits_{\varphi = 1}^{N_{\varphi}}\quad {\lambda_{\varphi}{i_{\varphi}}}}}} & (17)\end{matrix}$

where N is the number of stator phases. Thus, for a three-phase switchedreluctance motor, N_(φ)=3 (for the a, b, and c stator phases), while fora three phase PM motor N_(φ)=4 (for the three stator phases and theimaginary f rotor phase). Torque is found via coenergy by:$\begin{matrix}{T = {\frac{\partial\quad}{\partial\theta}\omega_{c}}} & (18)\end{matrix}$

The relationship between the abc-FoR and the αβ FoR is given by:$\begin{pmatrix}i_{\alpha} \\i_{\beta} \\i_{0}\end{pmatrix} = {\begin{pmatrix}1 & 0 & 1 \\\frac{- 1}{2} & {\frac{- 1}{2} \cdot \sqrt{3}} & 1 \\\frac{- 1}{2} & {\frac{1}{2} \cdot \sqrt{3}} & 1\end{pmatrix} \cdot \begin{pmatrix}i_{a} \\i_{b} \\i_{c}\end{pmatrix}}$

The derivatives of the abc FoR currents are related to the αβ-FoRcurrents by: $\begin{matrix}{{d\quad {\lambda_{a}\left( {i_{\alpha},i_{\beta},i_{f},\theta} \right)}} = {{\frac{\partial\quad}{\partial i_{\alpha}}{\lambda_{a} \cdot {di}_{\alpha}}} + {\frac{\partial\quad}{\partial i_{\beta}}{\lambda_{a} \cdot {di}_{\beta}}} + {\frac{\partial\quad}{\partial i_{f}}{\lambda_{a} \cdot {di}_{f}}}}} & (19) \\{{d\quad {\lambda_{b}\left( {i_{\alpha},i_{\beta},i_{f},\theta} \right)}} = {{\frac{\partial\quad}{\partial i_{\alpha}}{\lambda_{b} \cdot {di}_{\alpha}}} + {\frac{\partial\quad}{\partial i_{\beta}}{\lambda_{b} \cdot {di}_{\beta}}} + {\frac{\partial\quad}{\partial i_{f}}{\lambda_{b} \cdot {di}_{f}}}}} & (20) \\{{d\quad {\lambda_{c}\left( {i_{\alpha},i_{\beta},i_{f},\theta} \right)}} = {{\frac{\partial\quad}{\partial i_{\alpha}}{\lambda_{c} \cdot {di}_{\alpha}}} + {\frac{\partial\quad}{\partial i_{\beta}}{\lambda_{c} \cdot {di}_{\beta}}} + {\frac{\partial\quad}{\partial i_{f}}{\lambda_{c} \cdot {di}_{f}}}}} & (21) \\{{d\quad {\lambda_{f}\left( {i_{\alpha},i_{\beta},i_{f},\theta} \right)}} = {{\frac{\partial\quad}{\partial i_{\alpha}}{\lambda_{f} \cdot {di}_{\alpha}}} + {\frac{\partial\quad}{\partial i_{\beta}}{\lambda_{f} \cdot {di}_{\beta}}} + {\frac{\partial\quad}{\partial i_{f}}{\lambda_{f} \cdot {di}_{f}}}}} & (22)\end{matrix}$

and

di _(a) =di _(α)  (23) $\begin{matrix}{{di}_{b} = {{\frac{- 1}{2} \cdot {di}_{\alpha}} - {\frac{\sqrt{3}}{2} \cdot {di}_{\beta}}}} & (24) \\{{di}_{c} = {{\frac{- 1}{2} \cdot {di}_{\alpha}} + {\frac{\sqrt{3}}{2} \cdot {di}_{\beta}}}} & (25)\end{matrix}$

di _(f) =di _(f)  (25)

Since a switched reluctance machine has no permanent magnets, in thecase of a switched reluctance machine Equation (26) is as follows:

di_(f)≡0

The expression for coenergy is thus written in the αβ-FoR:$\begin{matrix}{\omega_{c} = {{\int{\lambda_{a}{i_{\alpha}}}} - {\frac{1}{2} \cdot {\int{\lambda_{b}{i_{\alpha}}}}} - {\frac{\sqrt{3}}{2} \cdot {\int{\lambda_{b}{i_{\beta}}}}} - {\frac{1}{2} \cdot {\int{\lambda_{c}{i_{\alpha}}}}} + {\frac{\sqrt{3}}{2} \cdot {\int{\lambda_{c}{i_{\beta}}}}} + {\int{\lambda_{f}{i_{f}}}}}} & (27)\end{matrix}$

For convenience, the individual integral components of this expressionare set equal to D₁, D₂, D₃, D₄, D₅ and D₆, respectively:$\begin{matrix}{\omega_{c} = {D_{1} - {\frac{1}{2} \cdot D_{2}} - {\frac{\sqrt{3}}{2} \cdot D_{3}} - {\frac{1}{2}D_{4}} + {\frac{\sqrt{3}}{2} \cdot D_{5}} + D_{6}}} & (28)\end{matrix}$

Again, in the particular case of a switched reluctance machine:

D₆≡0

For a switched reluctance machine, Equations (27) and (28) thereforereduce to: $\begin{matrix}{\omega_{c} = {{\int{\lambda_{a}{i_{\alpha}}}} - {\frac{1}{2} \cdot {\int{\lambda_{b}{i_{\alpha}}}}} - {\frac{\sqrt{3}}{2} \cdot {\int{\lambda_{b}{i_{\beta}}}}} - {\frac{1}{2} \cdot {\int{\lambda_{c}{i_{\alpha}}}}} + {\frac{\sqrt{3}}{2} \cdot {\int{\lambda_{c}{i_{\beta}}}}}}} & (29) \\{\omega_{c} = {D_{1} - {\frac{1}{2} \cdot D_{2}} - {\frac{\sqrt{3}}{2} \cdot D_{3}} - {\frac{1}{2}D_{4}} + {\frac{\sqrt{3}}{2} \cdot D_{5}}}} & (30)\end{matrix}$

Next, an integral path is selected. To this end, an integral pathconsisting of three directed line segments (DLS) is defined over whichto evaluate the integral giving coenergy. For the purposes of thisdisclosure, the dummy variable of integration is ξ while phase currentvariables (i_(a),i_(b),i_(c),i_(α),i_(β)) are in lower case. Theirassociated final values, with respect to path integrals, are(I_(a),I_(b),I_(c),I_(α),I_(β)).

Directed line segment one:

i_(α)=0 di_(α)=0

i_(β)=0 di_(β)=0

i_(f) is the variable of integration, ranging from 0 to I_(f).

Directed line segment two:

i_(α)=0 di_(α)=0

i_(β) is the variable of integration, ranging from 0 to I_(β)

i_(f)=I_(f) di_(f)=0

Directed line segment three:

i_(α) is the variable of integration, ranging from 0 to I_(α)

i_(β)=I_(β) di_(β)=0

i_(f)=I_(f) di_(f)=0

FIG. 3 illustrates an integration path defined as a sequence of directedline segments 51, 52, 53.

Each of the integrals D₁-D₆ are then evaluated over the selected path.Evaluating D₁:,

D ₁=∫λ_(a) di _(α)

This is identically zero over all but the third DLS, hence:$\begin{matrix}\begin{matrix}\begin{matrix}{D_{1} = {\int_{0}^{I_{\alpha}}{\sum\limits_{p = 0}^{P}\quad {\xi^{p} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{g_{apqrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{apqrn} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){\xi}}}}}}}}}}} \\{{{yielding}:}\quad}\end{matrix} \\{\quad {D_{1} = {\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{g_{apqrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {{h_{apqrn} \cdot \cos}\left( {n \cdot \theta} \right)}} \right)}}}}}}}}\quad}\end{matrix} & (31)\end{matrix}$

Evaluating D₂:

 D ₂=∫λ_(b) di _(α)

This is also identically zero over all but the third directed linesegment. $\begin{matrix}\begin{matrix}\begin{matrix}{D_{2} = {\int_{0}^{I_{\alpha}}{\sum\limits_{p = 0}^{P}\quad {\xi^{p} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{g_{bpqrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{bpqrn} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){\xi}}}}}}}}}}} \\{{{yielding}:}\quad}\end{matrix} \\{\quad {D_{2} = {\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{g_{bpqrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {{h_{bpqrn} \cdot \cos}\left( {n \cdot \theta} \right)}} \right)}}}}}}}}\quad}\end{matrix} & (32)\end{matrix}$

Evaluating D₃:

D ₃=∫λ_(b) di _(β)

This is identically zero over all but the second directed line segment,where all terms other than those where p=0 are zero: $\begin{matrix}\begin{matrix}\begin{matrix}{D_{3} = {\int_{0}^{I_{\beta}}{\sum\limits_{q = 0}^{Q}\quad {\xi^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{g_{b0qrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{b0qrn} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){\xi}}}}}}}}} \\{{{yielding}:}\quad}\end{matrix} \\{\quad {D_{3} = {\sum\limits_{q = 0}^{Q}\quad {\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{g_{b0qrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {{h_{b0qrn} \cdot \cos}\left( {n \cdot \theta} \right)}} \right)}}}}}}\quad}\end{matrix} & (33)\end{matrix}$

Evaluating D₄:

D ₄=∫λ_(c) di _(α)

This is identically zero over all but the third directed fine segment,where all terms are zero except for those where p=q=0: $\begin{matrix}\begin{matrix}\begin{matrix}{D_{4} = {\int_{0}^{I_{\alpha}}{\left\lbrack {\sum\limits_{p = 0}^{P}\quad {\xi^{p} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{g_{cpqrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{cpqrn} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}} \right\rbrack {\xi}}}} \\{{{yielding}:}\quad}\end{matrix} \\{\quad {D_{4} = {\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{g_{cpqrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {{h_{cpqrn} \cdot \cos}\left( {n \cdot \theta} \right)}} \right)}}}}}}}}\quad}\end{matrix} & (34)\end{matrix}$

Evaluating D₅:

D ₅=∫λ_(c) di _(β)

This is identically zero over all but the second directed line segment.$\begin{matrix}\begin{matrix}\begin{matrix}{D_{5} = {\int_{0}^{I_{\beta}}{\sum\limits_{q = 0}^{Q}\quad {\xi^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{g_{c0qrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{c0qrn} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){\xi}}}}}}}}} \\{{{yielding}:}\quad}\end{matrix} \\{\quad {D_{5} = {\sum\limits_{q = 0}^{Q}\quad {\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{g_{c0qrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {{h_{c0qrn} \cdot \cos}\left( {n \cdot \theta} \right)}} \right)}}}}}}\quad}\end{matrix} & (35)\end{matrix}$

Evaluating D₆:

D ₆=∫λ_(f) di _(f)

This term is identically zero over all but the first directed linesegment. $\begin{matrix}{{{D_{6} = {\int_{0}^{I_{f}}{\left\lbrack {\sum\limits_{r = 0}^{R}\quad {i_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{g_{f00rn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{f00rn} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}} \right\rbrack {\xi}}}}{{yielding}\text{:}}\quad {D_{6} = {\sum\limits_{r = 0}^{R}\quad {\frac{I_{f}^{r}}{r + 1} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{g_{f00rn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {{h_{f00rn} \cdot \cos}\left( {n \cdot \theta} \right)}} \right)}}}}}\quad} & (36)\end{matrix}$

Substituting for D₁, i=1, . . . ,6 into Equation (29): $\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{\omega_{c} = {{\left\lbrack {\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\left( {{g_{apqrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{apqrn} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}} \right\rbrack \ldots} +}} \\{{{\frac{- 1}{2} \cdot \left\lbrack {\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\left( {{g_{bpqrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{bpqrn} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}} \right\rbrack}\ldots} +}\end{matrix} \\{{{\frac{- \sqrt{3}}{2} \cdot \left\lbrack {\sum\limits_{q = 0}^{Q}{\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\left( {{g_{b0qrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{b0qrn} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}} \right\rbrack}\ldots} +}\end{matrix} \\{{{\frac{- 1}{2} \cdot \left\lbrack {\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\left( {{g_{cpqrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{cpqrn} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}} \right\rbrack}\ldots} +}\end{matrix} \\{{{\frac{\sqrt{3}}{2} \cdot \left\lbrack {\sum\limits_{q = 0}^{Q}\quad {\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\left( {{g_{c0qrn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{c0qrn} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}} \right\rbrack}\ldots} +} \\{\sum\limits_{r = 0}^{R}\quad {\frac{I_{f}^{r}}{r + 1} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{g_{f00rn} \cdot {\sin \left( {n \cdot \theta} \right)}} + {{h_{f00rn} \cdot \cos}\left( {n \cdot \theta} \right)}} \right)}}}\end{matrix} & (37)\end{matrix}$

Substituting Equation (37) into Equation (18): $\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{T = {{\left\lbrack {\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{apqrn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{apqrn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}} \right\rbrack \ldots} +}} \\{{{\frac{- 1}{2} \cdot \left\lbrack {\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{bpqrn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{bpqrn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}} \right\rbrack}\ldots} +}\end{matrix} \\{{{\frac{- \sqrt{3}}{2} \cdot \left\lbrack {\sum\limits_{q = 0}^{Q}{\frac{I_{\beta}^{p + 1}}{q + 1} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{b0qrn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{b0qrn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}} \right\rbrack}\ldots} +}\end{matrix} \\{{{\frac{- 1}{2} \cdot \left\lbrack {\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{cpqrn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{cpqrn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}} \right\rbrack}\ldots} +}\end{matrix} \\{{{\frac{\sqrt{3}}{2} \cdot \left\lbrack {\sum\limits_{q = 0}^{Q}\quad {\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{f}^{r} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{c0qrn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{c0qrn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}} \right\rbrack}\ldots} +} \\{\sum\limits_{r = 0}^{R}\quad {\frac{I_{f}^{r}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{f00rn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{f00rn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}\end{matrix} & (38)\end{matrix}$

Using the identities introduced earlier in Equations (12) and (13),Equation (38) is rewritten: $\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{T = {{\left\lbrack {\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{apqn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{apqn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}} \right\rbrack \ldots} +}} \\{{{\frac{- 1}{2} \cdot \left\lbrack {\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{bpqn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{bpqn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}} \right\rbrack}\ldots} +}\end{matrix} \\{{{\frac{- \sqrt{3}}{2} \cdot \left\lbrack {\sum\limits_{q = 0}^{Q}{\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{b0qn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{b0qn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}} \right\rbrack}\ldots} +}\end{matrix} \\{{{\frac{- 1}{2} \cdot \left\lbrack {\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{cpqn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{cpqn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}} \right\rbrack}\ldots} +}\end{matrix} \\{{{\frac{\sqrt{3}}{2} \cdot \left\lbrack {\sum\limits_{q = 0}^{Q}\quad {\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{c0qn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{c0qn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}} \right\rbrack}\ldots} +} \\{\sum\limits_{r = 0}^{R}\quad {\frac{I_{f}^{r}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{f00rn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{f00rn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}\end{matrix} & (38)\end{matrix}$

The electrical model parameters naturally flow from the expression forflux linkage to that for torque. However, for motors employing permanentmagnets, additional parameters appear as the electrical model istransformed into the torque model. Physically, these parameters relateto how the magnets on the motor interact with themselves—coggingparameters. It is not mathematically obvious how changes in motorterminal current and voltage, as the motor spins, indicate or measurethis behavior. In other words, the cogging parameters are unobservablesolely via feedback from the motor terminals as the rotor turns.

Various methods in accordance with the present invention are availableto deal with these unobservable parameters. For example, in oneembodiment, the torque model terms with unobservable parameters aregrouped together: $\begin{matrix}\begin{matrix}{T = \quad {{\sum\limits_{p = 0}^{P}\quad {\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}\quad {{n \cdot \begin{bmatrix}{{{\left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{\left( {{- H_{apqn}} + \frac{H_{bpqn}}{2} + \frac{H_{cpqn}}{2}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\end{bmatrix}}\ldots}}}}}} +}} \\{\quad {{\sum\limits_{q = 0}^{Q}\quad {\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}\quad {{\frac{\sqrt{3}}{2} \cdot n \cdot \left\lbrack {{\left( {{- G_{b0qn}} + G_{c0qn}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}} + {\left( {H_{b0qn} - H_{c0qn}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}} \right\rbrack}\quad \ldots}}}}\quad +}\quad} \\{\quad {\sum\limits_{r = 0}^{R}\quad {\frac{I_{f}^{r}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{f00rn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{f00rn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}\end{matrix} & (39)\end{matrix}$

For the particular case of a switched reluctance machine theseparameters are simply not present (no magnets in the motor construction)and Equation (39) reduces to: $\begin{matrix}\begin{matrix}{T = \quad {{\sum\limits_{p = 0}^{P}\quad {\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}\quad {{n \cdot \begin{bmatrix}{{\left( {g_{apqn} - \frac{g_{bpqn}}{2} - \frac{g_{cpqn}}{2}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} \\{{+ \left( {{- h_{apqn}} + \frac{h_{bpqn}}{2} + \frac{h_{cpqn}}{2}} \right)} \cdot {\sin \left( {n \cdot \theta} \right)}}\end{bmatrix}}\ldots}}}}}} +}} \\{\quad {\sum\limits_{q = 0}^{Q}\quad {\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \quad \frac{\sqrt{3}}{2} \cdot \left\lbrack {{\left( {{- g_{b0qn}} + g_{c0qn}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}} + {\left( {h_{b0qn} - h_{c0qn}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}} \right\rbrack}}}}\quad}\end{matrix} & (40)\end{matrix}$

The triple and first double summation terms in Equation (39) involvecoefficients that are observable through the motor terminals while thesecond double summation groups those terms that are not observable.

Cogging torque is measured, directly or indirectly, to provide data usedto derive an expression that can be substituted for the unknown groupedterms. For example, by spinning the motor on a test rig (no currentapplied to the motor windings) and measuring the cogging, a Fourierseries can be fitted directly to the test data and this known expressionis substituted for the unknown grouped terms. Consider Equation (39) inthe particular case when there no current is flowing in the statorphases:

I_(α)=I_(β)=0

The grouped unobservable terms from Equation (39) are as follows:$\sum\limits_{r = 0}^{R}\quad {\frac{I_{f}^{r}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}\quad \left( {{g_{f00rn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{f00rn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}$

Since I_(f) is nominally constant, the effect of cogging can be modeledby a trigonometric polynomial: $\begin{matrix}{\sum\limits_{n = 1}^{N}\quad \left( {{p_{n} \cdot {\sin \left( {n \cdot \theta} \right)}} + {q_{n} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)} & (41)\end{matrix}$

The p_(n) and q_(n) parameters are provided by the cogging torquemeasurement obtained from the unenergized motor. It is then assumed thatEquation (41) can replace the unobservable component of the expressionfor torque in Equation (39), that is:${\sum\limits_{r = 0}^{R}\quad {\frac{I_{f}^{r}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}\quad {n \cdot \left( {{g_{f00rn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{f00rn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}} = {\sum\limits_{n = 1}^{N}\quad \left( {{p_{k} \cdot {\sin \left( {n \cdot \theta} \right)}} + {q_{k} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}$

which yields: $\begin{matrix}\begin{matrix}{T = \quad {{\sum\limits_{p = 0}^{P}\quad {\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}\quad {{n \cdot \begin{bmatrix}{{{\left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{\left( {{- H_{apqn}} + \frac{H_{bpqn}}{2} + \frac{H_{cpqn}}{2}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\end{bmatrix}}\ldots}}}}}} +}} \\{\quad {{\sum\limits_{q = 0}^{Q}\quad {\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}\quad {{n \cdot \left\lbrack {{\left( {{\frac{- \sqrt{3}}{2} \cdot G_{b0qn}} + {\frac{\sqrt{3}}{2} \cdot G_{c0qn}}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}} + {\left( {{\frac{\sqrt{3}}{2} \cdot H_{b0qn}} - {\frac{\sqrt{3}}{2} \cdot H_{c0qn}}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}} \right\rbrack}\quad \ldots}}}}\quad +}\quad} \\{\quad {\sum\limits_{r = 1}^{N}\left( {{p_{k} \cdot {\sin \left( {n \cdot \theta} \right)}} + {q_{k} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}\end{matrix} & (42)\end{matrix}$

Other techniques in accordance with the invention use an indirectmeasurement of cogging torque. This generally involves three steps.First, during commissioning, the motor is spun at speed unloaded (freeshaft) at one or more speeds. Since there is no load applied to themotor shaft, the motor current is minimal. A simple linear voltage model(for example, including only phase currents and BEMF components—the αand β currents appear independent and to the first power only) of themotor is fitted from the voltage, current and shaft sensor information.The linear model will predict the operation of the motor at light loadsquite accurately. Next, the torque model for the motor is derived asdescribed previously from the linear voltage motor. The torque modelwill predict the output torque of the motor at light loads quiteaccurately.

In an alternative method, during commissioning, the motor is spun atspeed unenergized and the terminal voltage is measured. Using this data,part of the complete motor model, the BEMF component, can be fitted.Using the same method as that for the complete model, a torque model canbe created from this partial electrical model. The resultant model doesnot fully describe how the motor generates torque over the full currentrange, but for small values of current it is reasonably accurate.

The motor is then controlled so that position is maintained. That is,the controller energizes the windings until the motor holds a demandedposition, against the cogging torque (it is assumed that no other loadis applied—the motor is operating free shaft). Knowing a relativelyaccurate torque model for these low current ranges, the value of coggingtorque at any particular angle can now be calculated. In this way, byrepeating the above process over a full revolution, data can becollected concerning the cogging model. A Fourier series, for example,can be fitted, a look-up table can be created, etc., as previously andthis is used in place of those terms in the torque model that involvesunobservable parameters.

For numerical reasons it is desirable that the αβ-FoR currents arenormalized so that they lie in the closed interval [−1,1]. This processis best understood by considering Equation (1): $\begin{matrix}{\lambda_{\varphi} = {\sum\limits_{p = 0}^{P}\quad {i_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}\quad {i_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {i_{f}^{r} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{g_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}}} & (1)\end{matrix}$

Assuming flux is directly measured and that measured currents arenormalized using some scale factor I, then Equation (1) becomes:$\begin{matrix}{\lambda_{\varphi} = {\sum\limits_{p = 0}^{P}\quad {\left( \frac{i_{\alpha}}{I} \right)^{p} \cdot {\sum\limits_{q = 0}^{Q}\quad {\left( \frac{i_{\beta}}{I} \right)^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {\left( \frac{i_{f}}{I} \right)^{r} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{g_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}}} & (43)\end{matrix}$

The corresponding voltage equation is given by: $\begin{matrix}\begin{matrix}{v_{\varphi} = \quad {R_{\varphi} + {\frac{i_{\alpha}}{I} \cdot R_{\varphi \quad \alpha}} + {\frac{i_{\beta}}{I} \cdot R_{\varphi \quad \beta}} + {{\frac{i_{\alpha}}{I} \cdot \frac{i_{\beta}}{I} \cdot R_{\varphi \quad \alpha \quad \beta}}\quad \ldots} +}} \\{\quad {{\sum\limits_{p = 1}^{P}{\left\lbrack {p\quad {\left( \frac{i_{\alpha}}{I} \right)^{p - 1} \cdot \frac{\quad}{t}}\quad \left( \frac{i_{\alpha}}{I} \right)} \right\rbrack \cdot {\sum\limits_{q = 0}^{Q}\quad {\left( \frac{i_{\beta}}{I} \right)^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {\left( \frac{i_{f}}{I} \right)^{r} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{g_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}\quad +}} \\{\quad {{\sum\limits_{p = 0}^{P}\quad {\left( \frac{i_{\alpha}}{I} \right)^{p} \cdot {\sum\limits_{q = 1}^{Q}\quad {\left\lbrack {q\quad {\left( \frac{i_{\beta}}{I} \right)^{q - 1} \cdot \frac{\quad}{t}}\quad \left( \frac{i_{\beta}}{I} \right)} \right\rbrack {\sum\limits_{r = 0}^{R}\quad {\left( \frac{i_{f}}{I} \right)^{r} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{g_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}} +}} \\{\quad {\sum\limits_{p = 0}^{P}\quad {\left( \frac{i_{\alpha}}{I} \right)^{p} \cdot {\sum\limits_{q = 0}^{Q}\quad {\left( \frac{i_{\beta}}{I} \right)^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {\left( \frac{i_{f}}{I} \right)^{r} \cdot {\sum\limits_{n = 1}^{N}{\omega \cdot n \cdot \quad \left( {{g_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}\end{matrix} & (44)\end{matrix}$

Terms such as $\frac{\quad}{t}\left( \frac{i_{\alpha}}{I} \right)$

correspond to the derivative of the normalized current. Two furtherequations of interest referenced earlier are those relating flux tocoenergy and coenergy to torque: $\begin{matrix}{\omega_{c} = {\int{\sum\limits_{\varphi = 1}^{N_{\varphi}}{\lambda_{\varphi}{i_{\varphi}}}}}} & (17) \\{T = {\frac{\partial\quad}{\partial\theta}\omega_{c}}} & (18)\end{matrix}$

The integral in Equation (17) is with respect to true currentmeasurement. From this, if the torque equation is evaluated usingnormalized current then true torque value is obtained my multiplying bythe scaling factor I.

The torque model is used by the solver 34 to calculate appropriatecurrents for smooth torque with zero angle sensitivity. Over a singlerevolution of the shaft, the torque levels calculated may be such as toreject some disturbance. Further, as discussed above, errors in positionmeasurement degrade performance, and the performance tends to rapidlydeteriorate with errors in angular measurement. To achieve smooth torquewith zero, or at least reduced sensitivity to angular measurement, asolution is derived that considers the change in torque with respect toangle, then minimizes this variance. First, a set of nonlinear equationsare solved. For N functional relationships involving variables x_(i),i=1, . . . ,N:

F _(i)(x ₁ ,x ₂ , . . . ,x _(N))=0 i=1, . . . ,N  (45)

Adopting vector notation and expanding the functions F_(i) using theTaylor series: $\begin{matrix}{{F_{i}\left( {x + {\delta \quad x}} \right)} = {{F_{i}(x)} + {\sum\limits_{j = 1}^{N}{\frac{\partial\quad}{\partial x_{j}}{F_{i} \cdot \delta}\quad x_{j}}} + {O\left( {\delta \quad x^{2}} \right)}}} & (46)\end{matrix}$

The Jacobian is written: $\begin{matrix}{J_{ij} = {\frac{\partial\quad}{\partial x_{j}}F_{i}}} & (47)\end{matrix}$

Then:

F(x+δx)=F(x)+J·δx+O(δx ²)  (48)

From this, a set of linear equations is obtained:

 J·δx _(new) =−F  (49)

Solving the linear equation above yields:

x _(new) =x _(old) +δx _(new)  (50)

In this particular case, the non-linear equations are for torque andzero sensitivity with respect to angle condition.

In the problem under discussion, while the Jacobian matrix willgenerally be small (two by two or three by three) there is littlecomputational cost in calculating the inverse. However, there issignificant overhead in calculating the individual elements of theJacobian. One way to reduce the cost of computing is to keep theJacobian constant for some number of iterations p>1. Such a technique(cyclic updating of the Jacobian matrix) offsets the reduction incomputational cost with a deterioration in convergence rate for thesolution. There are also multi-dimensional secant-type methods thatavoid the explicit calculation of the Jacobian through the use ofmulti-dimensional finite differencing.

With the method described above, the initial guess to the solution hasto be reasonably close because global convergence is not guaranteed.This typically is not problematical while searching for solutions for PMmotors but may be so for switched reluctance motors. One possiblesolution is to examine the use of quasi-Newton methods, which possessglobal convergence properties, possibly with a restart scheme.

Torque may be calculated via coenergy using Equation (39), repeated asfollows: $\begin{matrix}{T = {{\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}{I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \begin{bmatrix}{{{\left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{\left( {{- H_{apqn}} + \frac{H_{bpqn}}{2} + \frac{H_{cpqn}}{2}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\end{bmatrix}}\ldots}}}}}} + {\sum\limits_{q = 0}^{Q}{\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{{\frac{\sqrt{3}}{2} \cdot n \cdot \left\lbrack {{\left( {{- G_{b0qn}} + G_{c0qn}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}} + {\left( {H_{b0qn} - H_{c0qn}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}} \right\rbrack}\quad \ldots}}}} + {\sum\limits_{r = 0}^{R}{\frac{I_{f}^{r}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{f00rn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{f00rn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}} & (39)\end{matrix}$

Ignoring the unobservable parameters and re-arranging Equation (39)yields: $\begin{matrix}{0 = {{\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}{I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \begin{bmatrix}{{{\left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{\left( {{- H_{apqn}} + \frac{H_{bpqn}}{2} + \frac{H_{cpqn}}{2}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\end{bmatrix}}\quad \ldots}}}}}} + {\sum\limits_{q = 0}^{Q}{\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{\frac{\sqrt{3}}{2} \cdot n \cdot \left\lbrack {{\left( {{- G_{b0qn}} + G_{c0qn}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}} + {{\left( {H_{b0qn} - H_{c0qn}} \right) \cdot \sin}\left( {n \cdot \theta} \right)}} \right\rbrack}}}} - T}} & (51)\end{matrix}$

The necessary partial derivatives or entries to the Jacobian, withrespect to i_(α) and i_(β), can now be derived. $\begin{matrix}{{\frac{\partial\quad}{\partial i_{\alpha}}{T\left( {i_{\alpha},i_{\beta},\theta} \right)}} = {\sum\limits_{p = 0}^{P}{I_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \begin{bmatrix}{{{\left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{\left( {{- H_{apqn}} + \frac{H_{bpqn}}{2} + \frac{H_{cpqn}}{2}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\end{bmatrix}}}}}}}} & (52) \\{{{\frac{\partial\quad}{\partial i_{\beta}}{T\left( {i_{\alpha},i_{\beta},\theta} \right)}} = {{\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}{{qI}_{\beta}^{q - 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \begin{bmatrix}{{{\left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{\left( {{- H_{apqn}} + \frac{H_{bpqn}}{2} + \frac{H_{cpqn}}{2}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\end{bmatrix}}\quad \ldots}}}}}} + {\sum\limits_{q = 0}^{Q}{I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{\frac{\sqrt{3}}{2} \cdot n \cdot \begin{bmatrix}{{{\left( {{- G_{b0qn}} + G_{c0qn}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{\left( {{- H_{b0qn}} - H_{c0qn}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\end{bmatrix}}}}}}}{{Hence}:}} & (53) \\{{J_{11} = {\sum\limits_{p = 0}^{P}{I_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \begin{bmatrix}{{{\left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{\left( {{- H_{apqn}} + \frac{H_{bpqn}}{2} + \frac{H_{cpqn}}{2}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\end{bmatrix}}}}}}}}\quad} & (54) \\{J_{12} = {{\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}{{qI}_{\beta}^{q - 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \begin{bmatrix}{{{\left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{\left( {{- H_{apqn}} + \frac{H_{bpqn}}{2} + \frac{H_{cpqn}}{2}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\end{bmatrix}}\quad \ldots}}}}}} + {\sum\limits_{q = 0}^{Q}{I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{\frac{\sqrt{3}}{2} \cdot n \cdot \left\lbrack {{\left( {{- G_{b0qn}} + G_{c0qn}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}} + {{\left( {H_{b0qn} - H_{c0qn}} \right) \cdot \sin}\left( {n \cdot \theta} \right)}} \right\rbrack}}}}}} & (55)\end{matrix}$

Regarding zero angular sensitivity, sensitivity with respect to angle isgiven by: $\begin{matrix}{\frac{\partial\quad}{\partial\theta}T} & (56)\end{matrix}$

From Equation (39): $\begin{matrix}{{\frac{\partial\quad}{\partial\theta}T} = {{\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}{I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{{n^{2} \cdot \left\lbrack \quad \begin{matrix}{{{- \left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right)} \cdot {\sin \left( {n \cdot \theta} \right)}}\quad \ldots \text{+}} \\{\left( {{- H_{apqn}} + {\frac{H_{bpqn}}{2}\text{+}\frac{H_{cpqn}}{2}}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\end{matrix}\quad \right\rbrack}\quad \ldots}}}}}} + \quad {\sum\limits_{q = 0}^{Q}{\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{{n^{2} \cdot \frac{\sqrt{3}}{2} \cdot \begin{bmatrix}{{{{- \left( {{- G_{b0qn}} + G_{c0qn}} \right)} \cdot {\sin \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{\left( {H_{b0qn} - H_{c0qn}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\end{bmatrix}}{{Hence}:}}}}}}} & (57) \\{{J_{21} = {\frac{\partial\quad}{\partial i_{\alpha}}\frac{\partial\quad}{\partial\theta}T}}{{and}:}} & (58) \\{J_{22} = {\frac{\partial\quad}{\partial i_{\beta}}\frac{\partial\quad}{\partial\theta}T}} & (59)\end{matrix}$

By changing the order of the partial derivatives, in Equations (58) and(59): $\begin{matrix}{J_{21} = {\frac{\partial\quad}{\partial\theta}J_{11}}} & (60) \\{J_{22} = {\frac{\partial\quad}{\partial\theta}J_{12}}} & (61)\end{matrix}$

As noted above, cogging torque cannot be estimated using data availablesolely from the motor terminals. The cogging model discussed above canbe incorporated directly into the solver 34. Recall that the observableand unobservable torque model terms were grouped together in Equation(39). If total motor torque is separated into torque estimated viaterminal variables (T_(tv)) and cogging torque (T_(cog)), then:

T=T _(tv) +T _(cog)  (62)

and $\begin{matrix}{{\frac{\partial\quad}{\partial\theta}T} = {{\frac{\partial\quad}{\partial\theta}T_{tv}} + {\frac{\partial\quad}{\partial\theta}T_{cog}}}} & (63)\end{matrix}$

Considering T_(cog) as solely a function of angle (representing theangle dependant but current independent cogging) in terms of a Fourierseries: $\begin{matrix}{{\frac{\partial\quad}{\partial i_{\alpha}}T_{cog}} = {{\frac{\partial\quad}{\partial i_{\beta}}T_{cog}} = 0}} & (64) \\{{\frac{\partial\quad}{\partial i_{\alpha}}\left( {\frac{\partial\quad}{\partial\theta}T_{cog}} \right)} = {{\frac{\partial\quad}{\partial i_{\beta}}\left( {\frac{\partial\quad}{\partial\theta}T_{cog}} \right)} = 0}} & (65)\end{matrix}$

From Equations (64) and (65) it is seen that there is no effect upon thecalculation of the Jacobian matrix, see Equation (39) and Equations (51)to (61). The only effect is upon the calculation of true torque (T) andits partial derivative with respect to angle, or the sensitivity. RecallEquation (39): $\begin{matrix}{T = \quad {{\sum\limits_{p = 0}^{P}\quad {\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}\quad {{n \cdot \begin{bmatrix}{{\left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} \\{{+ \left( {{- H_{apqn}} + \frac{H_{bpqn}}{2} + \frac{H_{cpqn}}{2}} \right)} \cdot {\sin \left( {n \cdot \theta} \right)}}\end{bmatrix}}\ldots}}}}}} +}} \\{\quad {{\sum\limits_{q = 0}^{Q}\quad {\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}\quad {n{\frac{\sqrt{3}}{2} \cdot \left\lbrack {{\left( {{- G_{b0qn}} + G_{c0qn}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}} + {\left( {H_{b0qn} - H_{c0qn}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}} \right\rbrack}\quad \ldots}}}}\quad +}\quad} \\{\quad {\sum\limits_{r = 0}^{R}\quad {\frac{I_{f}^{r}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{f00rn} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{f00rn} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}\end{matrix}$

If, as mentioned previously, T_(cog) is treated as a function ofmechanical angle via a Fourier series: $\begin{matrix}{{T_{cog}(\theta)} = {\sum\limits_{n = 1}^{N_{c}}\quad \left( {{a_{n} \cdot {\sin \left( {n \cdot \theta} \right)}} + {b_{n} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}} & (66)\end{matrix}$

then Equation (39) yields: $\begin{matrix}\begin{matrix}{T = \quad {{\sum\limits_{p = 0}^{P}\quad {\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}\quad {{n \cdot \quad \begin{bmatrix}{{{\left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{\left( {{- H_{apqn}} + \frac{H_{bpqn}}{2} + \frac{H_{cpqn}}{2}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\end{bmatrix}}\ldots}}}}}} +}} \\{\quad {{\sum\limits_{q = 0}^{Q}\quad {\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}\quad {{\frac{\sqrt{3}}{2} \cdot n \cdot \left\lbrack {{\left( {{- G_{b0qn}} + G_{c0qn}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}} + {\left( {H_{b0qn} - H_{c0qn}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}} \right\rbrack}\quad \ldots}}}}\quad +}\quad} \\{\quad {\sum\limits_{n = 1}^{N_{c}}\left( {{a_{n} \cdot {\sin \left( {n \cdot \theta} \right)}} + {b_{n} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}\end{matrix} & (67)\end{matrix}$

True sensitivity is then the partial derivative with respect to angle ofthe expression for torque presented in Equation (67), see Equation (57):$\begin{matrix}\begin{matrix}{S_{a}\quad = {{\sum\limits_{p = 0}^{P}\quad {\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}\quad {{n^{2} \cdot \begin{bmatrix}{{{- \left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right)} \cdot {\sin \left( {n \cdot \theta} \right)}}\quad \ldots} \\{{+ \left( {{- H_{apqn}} + \frac{H_{bpqn}}{2} + \frac{H_{cpqn}}{2}} \right)} \cdot {\cos \left( {n \cdot \theta} \right)}}\end{bmatrix}}\ldots}}}}}} +}} \\{\quad {{\sum\limits_{q = 0}^{Q}\quad {\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}\quad {{\frac{\sqrt{3} \cdot n^{2}}{2} \cdot \left\lbrack {{{- \left( {{- G_{b0qn}} + G_{c0qn}} \right)} \cdot {\sin \left( {n \cdot \theta} \right)}} + {\left( {H_{b0qn} - H_{c0qn}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}} \right\rbrack}\quad \ldots}}}}\quad +}\quad} \\{\quad {\sum\limits_{n = 1}^{N_{c}}{n \cdot \left( {{a_{n} \cdot {\cos \left( {n \cdot \theta} \right)}} - {b_{n} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}\end{matrix} & (68)\end{matrix}$

To this point it has been assumed that the smooth torque solution isachieved in a point by point manner. An alternative approach is tocalculate the solution across the angle intervals in the interval (0,2π)simultaneously. For convenience, the following nomenclature isintroduced. Suppose for a particular torque and sensitivity demand thesolution is to be calculated at various angles:

θ(k) for all k=1, . . . ,N

The current column vector is given by: $\begin{matrix}{I = \begin{pmatrix}{i_{\alpha}\left( {\theta (1)} \right)} \\{i_{\alpha}\left( {\theta (2)} \right)} \\\cdots \\{i_{\alpha}\left( {\theta (N)} \right)} \\{i_{\beta}\left( {\theta (1)} \right)} \\{i_{\beta}\left( {\theta (2)} \right)} \\\cdots \\{i_{\beta}\left( {\theta (N)} \right)}\end{pmatrix}} & (69)\end{matrix}$

If the new

I(n+1)=I(n)+Δ(n)  (70)

The k^(th) torque vector is a row vector defined by:

φ_(Tk)=(0 . . . 0T(θ(k),i _(α)(k),i_(β)(k))0 . . . 0)

Similarly, the k'th sensitivity vector is defined by:

φ_(Sk)=(0 . . . 0S(θ(k),i _(α)(k),i _(β)(k))0 . . . 0)

These vectors can be stacked to form diagonal matrices: $\begin{matrix}{A = \begin{pmatrix}\varphi_{T1} \\\cdots \\\varphi_{TN}\end{pmatrix}} & {B = \begin{pmatrix}\varphi_{S1} \\\cdots \\\varphi_{SN}\end{pmatrix}}\end{matrix}$

Finally, appropriate partial derivatives with respect to the currentsare taken and the resultant matrices aggregated to form a 2N by 2Nmatrix: $\begin{matrix}{\Phi = \begin{pmatrix}{\frac{\partial\quad}{\partial i_{\alpha}}A\frac{\partial\quad}{\partial i_{\beta}}A} \\{\frac{\partial\quad}{\partial i_{\alpha}}B\frac{\partial\quad}{\partial i_{\beta}}B}\end{pmatrix}} & (71)\end{matrix}$

The desired torque and sensitivity at a particular angle θ(k) are givenby:

T(θ(k)) S(θ(k))

The 2N×1 demand vector D of these values over the angle range is givenby: $\begin{matrix}{D = \begin{pmatrix}{T_{d}\left( {\theta (1)} \right)} \\{T_{d}\left( {\theta (2)} \right)} \\\cdots \\{T_{d}\left( {\theta (N)} \right)} \\{S_{d}\left( {\theta (1)} \right)} \\\cdots \\{S_{d}\left( {\theta (1)} \right)}\end{pmatrix}} & (72)\end{matrix}$

and the actual values of torque and sensitivity resultant from anycurrent combination (i_(α),i_(β)) which constitute the iterated solutionare given by the column vector: $\begin{matrix}{A = \begin{pmatrix}{T\left( {{\theta (1)},{i_{\alpha}(1)},{i_{\beta}(1)}} \right)} \\{T\left( {{\theta (2)},{i_{\alpha}(2)},{i_{\beta}(2)}} \right)} \\\cdots \\{T\left( {{\theta (N)},{i_{\alpha}(N)},{i_{\beta}(N)}} \right)} \\{S\left( {{\theta (1)},{i_{\alpha}(1)},{i_{\beta}(1)}} \right)} \\\cdots \\{S\left( {{\theta (N)},{i_{\alpha}(N)},{i_{\beta}(N)}} \right)}\end{pmatrix}} & (73)\end{matrix}$

Using this notation and from the development initially presented:

Δ(n)=Φ⁻¹·(D−A)  (74)

with:

I(n+1)=I(n)+Δ(n)

Starting from a reasonable guess, three to fifteen iterations istypically sufficient when dealing with a PM motor. As in the switchedreluctance motor case previously, issues of convergence becomescritical. Typical seed values for the PM motor are usually chosen to lieupon a sine wave generated with some harmonic appropriate to the machinein question. Hence, in the unstacked version of the solver:$\begin{pmatrix}i_{\alpha} \\i_{\beta}\end{pmatrix} = \begin{pmatrix}0.2 \\{- 0.2}\end{pmatrix}$

In the stacked version this expression is replaced with sine feeds ofthe appropriate magnitude and harmonic, for example, five in a 12-10 PMmotor.

An alternative to approach to that outlined above is to view the problemas a constrained non-linear optimization task with respect to the alphabeta currents for each angle:${\frac{\partial\quad}{\partial\theta}{T\left( {i_{\alpha},i_{\beta},{\theta (k)}} \right)}} = 0$

T(i _(α) ,i _(β),θ(k))=T _(demand)

 MIN(i _(α) ² +i _(β) ²)

Such a problem is a particular case of that described in Sensitivity ofAutomatic Control Systems by Rosenwasser and Yusupov (CRC Press, 1999).

In some cases where cogging frequencies are high, the resultant currentprofiles may prove difficult to follow. Of course, increasingly complexand expensive drives can provide better current profile followingcapabilities. In practice, a reasonable balance between cost and currentfollowing capabilities is required. One solution is the removal of thehigher harmonic terms from all the sensitivity expressions presentedthroughout this disclosure. In particular, consider the expressionpresented in Equation (71). If the higher harmonic terms are ignored,above N_(t), then: $\begin{matrix}{S_{a} = {{\sum\limits_{p = 0}^{P}{\frac{I_{\alpha}^{p + 1}}{p + 1} \cdot {\sum\limits_{q = 0}^{Q}{I_{\beta}^{q} \cdot {\sum\limits_{n = 1}^{N}{{n^{2} \cdot \begin{bmatrix}{{{{- \left( {G_{apqn} - \frac{G_{bpqn}}{2} - \frac{G_{cpqn}}{2}} \right)} \cdot {\sin \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{\left( {{- H_{apqn}} + \frac{H_{bpqn}}{2} + \frac{H_{cpqn}}{2}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\end{bmatrix}}\ldots}}}}}} + {\sum\limits_{q = 0}^{Q}{\frac{I_{\beta}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{{\frac{\sqrt{3} \cdot n^{2}}{2} \cdot \left\lbrack {{{- \left( {{- G_{b0qn}} + G_{c0qn}} \right)} \cdot {\sin \left( {n \cdot \theta} \right)}} + {\left( {H_{b0qn} - H_{c0qn}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}} \right\rbrack}\quad \ldots}}}} + {\sum\limits_{n = 1}^{N_{t}}{n \cdot \left( {{a_{n} \cdot {\cos \left( {n \cdot \theta} \right)}} - {b_{n} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}} & (75)\end{matrix}$

Typically the cut-off harmonic will be in the range 35 to 40. This willresult in smoother current profiles, which in the presence of error inangular precision, will introduce higher frequency torque ripplesomewhat more quickly than would otherwise be the case.

If balanced feed is not used, it is necessary to use the abc-FoR. Aswith the αβ-FoR discussion above, the following disclosure is generallyprovided in terms of a three phase hybrid motor, though the model formcan be generalized into different types of rotating machines having anynumber of phases by one skilled in the art having the benefit of thisdisclosure. For notational convenience, the abc-stator phases arerepresented by “1,” “2” and “3” subscripts, while the single rotor phaseis represented by a “4” subscript.

The general form of the flux linkage model in the abc-FoR is given by:$\begin{matrix}{{\lambda_{\varphi}\left( {i_{1},i_{2},i_{3},i_{4},\theta} \right)} = {\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}} & (76)\end{matrix}$

Equation (4) provides the known equation: $\begin{matrix}{v_{\varphi} = {{i_{\varphi} \cdot R_{\varphi}} + {\frac{\quad}{t}\lambda_{\varphi}}}} & (4)\end{matrix}$

From Equation (76): $\begin{matrix}{{\frac{\quad}{t}\lambda_{\varphi}} = {{\sum\limits_{p = 0}^{P}{p \cdot i_{1}^{p - 1} \cdot \left( {\frac{\quad}{t}i_{1}} \right) \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{{qi}_{2}^{q - 1} \cdot \left( {\frac{\quad}{t}i_{2}} \right) \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{{i_{2}^{q} \cdot \underset{r = 0}{\overset{R}{\sum{r \cdot}}}}{i_{3}^{r - 1} \cdot \left( {\frac{\quad}{t}i_{3}} \right) \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{{si}_{4}^{s - 1} \cdot \left( {\frac{\quad}{t}i_{4}} \right) \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\omega \cdot n \cdot \left( {{g_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}}}} & (77)\end{matrix}$

Substituting Equation (77) into Equation (4): $\begin{matrix}{{v_{\varphi} = {{i_{\varphi} \cdot R_{\varphi}} + {\sum\limits_{p = 0}^{P}{p \cdot i_{1}^{p - 1} \cdot \left( {\frac{\quad}{t}i_{1}} \right) \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}}} + {\sum\limits_{p = 0}^{P}{{i_{1}^{p} \cdot \underset{q = 0}{\overset{Q}{\sum q}}}{i_{2}^{q - 1} \cdot \left( {\frac{\quad}{t}i_{2}} \right) \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{{i_{2}^{q} \cdot \underset{r = 0}{\overset{R}{\sum{r \cdot}}}}{i_{3}^{r - 1} \cdot \left( {\frac{\quad}{t}i_{3}} \right) \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{{si}_{4}^{s - 1} \cdot \left( {\frac{\quad}{t}i_{4}} \right) \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\omega \cdot n \cdot \left( {{g_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}}}}\quad} & (78)\end{matrix}$

As noted herein above, in the particular case of a switched reluctancemotor:

i_(f)≡0 and $\begin{matrix}{{\frac{\quad}{t}i_{f}} \equiv 0} & (9)\end{matrix}$

So in the case of a switched reluctance motor, Equation (78) yields$\begin{matrix}{{v_{\varphi} = {{i_{\varphi} \cdot R_{\varphi}} + {\sum\limits_{p = 0}^{P}{p \cdot i_{1}^{p - 1} \cdot \left( {\frac{\quad}{t}i_{1}} \right) \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}} + {\sum\limits_{p = 0}^{P}{{i_{1}^{p} \cdot \underset{q = 0}{\overset{Q}{\sum q}}}{i_{2}^{q - 1} \cdot \left( {\frac{\quad}{t}i_{2}} \right) \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{{i_{2}^{q} \cdot \underset{r = 0}{\overset{R}{\sum{r \cdot}}}}{i_{3}^{r - 1} \cdot \left( {\frac{\quad}{t}i_{3}} \right) \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}} + {\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{n = 1}^{N}{\omega \cdot n \cdot \left( {{g_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}}\quad} & (79)\end{matrix}$

Recall from Equations (17) and (18), coenergy and torque are found usingthe expressions: $\begin{matrix}{\omega_{c} = {\int{\sum\limits_{\varphi = 1}^{4}{\lambda_{\varphi}{i_{\varphi}}}}}} & (17) \\{T = {\frac{\partial\quad}{\partial\theta}\omega_{c}}} & (18)\end{matrix}$

Any reasonable integral path can be selected; in certain embodimentsfour directed line segments are used:

Variable of integration i₄ di₄≠C i₁,i₂,i₃,di₁,di₂,di₃=C

Variable of integration i₁ di₁≠C i₄=I₄ i₂,i₃,di₂,di₃,di₄=C

Variable of integration i₂ di₂≠C i₄=I₄ i₁=I₁ i₃,di₁,di₃,di₄=C

Variable of integration i₃ di₃≠C i₄=I₄ i₁=I₁ i₂=I₂ di₁,di₂,di₄=C

This path is selected to minimize the number of unobservable parametersappearing in the final expression for torque.

The integrals are then evaluated over the selected path. SubstitutingEquation (76) into Equation (17): $\begin{matrix}{\omega_{c} = {\int{\left\lbrack {\sum\limits_{\varphi = 1}^{4}{\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot \quad {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}} \right\rbrack {i_{\varphi}}}}} & (80)\end{matrix}$

For notational convenience, this integral is re-written: $\begin{matrix}{\omega_{c} = {\sum\limits_{\varphi = 1}^{4}F_{\varphi}}} & (81)\end{matrix}$

where, for φ=1, . . . ,4: $\begin{matrix}{F_{\varphi} = {\int{\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){i_{\varphi}}}}}}}}}}}}}} & (82)\end{matrix}$

In the particular case of a switched reluctance machine (no permanentmagnets) there is no associated stator phase: $\begin{matrix}{{F_{\varphi} = {{\int{\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){i_{\varphi}}\quad \varphi}}}}}}}}} = 1}},2,3.} & \quad\end{matrix}$

Integral One (F₁):$F_{1} = {\int{\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}\quad {i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{1{pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{1{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){i_{1}}}}}}}}}}}}}$

This is identically zero over all but the second DLS, where:

i₄=I₄ i₁,i₂,i₃=C di₂,d₃,d₄=C

hence: $\begin{matrix}{{F_{1} = {\int_{0}^{I_{1}}{\sum\limits_{p = 0}^{P}{\xi^{p} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{1{p00sn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{1{p00sn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){\xi}}}}}}}}}{{yielding}:}} & \quad \\{F_{1} = {\sum\limits_{p = 0}^{P}{\frac{I_{1}^{p + 1}}{p + 1} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}\left( {{g_{1{p00sn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{1{p00sn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}} & (83)\end{matrix}$

Integral Two (F₂):$F_{2} = {\int{\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{2{pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{2{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){i_{2}}}}}}}}}}}}}$

This is identically zero along all but the third DLS, where:

i₄=I₄ i_(i)=I₁ i₁,i₃=C di₁,di₃,di₄=C

hence: $\begin{matrix}{{F_{2} = {\int_{0}^{I_{2}}{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{\xi^{q} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{2{pq0sn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{2{pq0sn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){\xi}}}}}}}}}}}{{yielding}:}} & \quad \\{F_{2} = {\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{\frac{I_{2}^{q + 1}}{q + 1} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}\left( {{g_{2{pq0sn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{2{pq0sn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}}} & (84)\end{matrix}$

Integral Three (F₃):$F_{3} = {\int{\sum\limits_{p = 0}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{3{pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){i_{3}}}}}}}}}}}}}$

This is identically zero along all but the fourth DLS, where:

i₄=I₄ i_(i)=I₁ i₂=I₂ di₁,di₂,di₄=C

hence: $\begin{matrix}{{F_{3} = {\int_{0}^{I_{3}}{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{\xi^{r} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{3{pqsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){\xi}}}}}}}}}}}}}{{yielding}:}} & \quad \\{F_{3} = {\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}\left( {{g_{3{pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}} & (85)\end{matrix}$

Integral Four (F₄):$F_{4} = {\int{\sum\limits_{p = 0}^{P}{i_{1^{p}} \cdot {\sum\limits_{q = 0}^{Q}{i_{2^{q}} \cdot {\sum\limits_{r = 0}^{R}{i_{3^{r}} \cdot {\sum\limits_{s = 0}^{S}{i_{4^{s}} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{4{pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{4{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){i_{4}}}}}}}}}}}}}$

This is identically zero along all but the first DLS, where:

i₁,i₂,i₃,di₁,di₂,di₃=C $\begin{matrix}{{F_{4} = {\int_{0}^{I_{f}}{\sum\limits_{s = 0}^{S}{\xi^{s} \cdot {\sum\limits_{n = 1}^{N}\quad {\left( {{{g_{4000{sn}} \cdot \sin}\quad \left( {n \cdot \theta} \right)} + {h_{4000{sn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right){\xi}}}}}}}{{yielding}\text{:}}} & \quad \\{F_{4} = {\sum\limits_{s = 0}^{S}{\frac{I_{f}^{s + 1}}{s + 1} \cdot {\sum\limits_{n = 1}^{N}\left( {{{g_{4000{sn}} \cdot \sin}\quad \left( {n \cdot \theta} \right)} + {h_{4000{sn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}} & (86)\end{matrix}$

Substituting Equations (83) to (86) into Equation (81) results in theexpression: $\begin{matrix}{\omega_{c} = {{\sum\limits_{p = 0}^{P}{\frac{I_{1}^{p + 1}}{p + 1} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{1{p00sn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{1{p00sn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}\quad + {\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{\frac{I_{2}^{q + 1}}{q + 1} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{2{pq0sn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{2{pq0sn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}} + {\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{3{pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}}} + {\sum\limits_{s = 0}^{S}{\frac{I_{f}^{s + 1}}{s + 1} \cdot {\sum\limits_{n = 1}^{N}\left( {{g_{4000{sn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{4000{sn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}} & (87)\end{matrix}$

Recalling Equation (18) it is seen that, in the abc-FoR, torque is givenby: $\begin{matrix}{T = {{\sum\limits_{p = 0}^{P}{\frac{I_{1}^{p + 1}}{p + 1} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{g_{1{p00sn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{1{p00sn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}} + {\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{\frac{I_{2}^{q + 1}}{q + 1} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{g_{2{pq0sn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{2{pq0sn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}}}}\quad + {\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{g_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{3{pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}}}}}}\quad + {\sum\limits_{s = 0}^{S}{\frac{I_{f}^{s + 1}}{s + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{4000{sn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{4000{sn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}} & (88)\end{matrix}$

In the case of a switched reluctance motor: $\begin{matrix}{T = {{\sum\limits_{p = 0}^{P}{\frac{I_{1}^{p + 1}}{p + 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{g_{1{p00n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{1{p00n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}} + {\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{\frac{I_{2}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{g_{2{pq0n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{2{pq0n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}} + {\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{3{pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{3{pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}} & (89)\end{matrix}$

Data collection schemes suitable for parameter estimation were addressedabove, including exemplary constant phase current and varying phasecurrent schemes. The varying current scheme, which is typical ofpractical applications, is considered below. Recall Equation (78):$\begin{matrix}{v_{\varphi} = {{i_{\varphi} \cdot R_{\varphi}} + {\sum\limits_{p = 1}^{P}{p \cdot i_{1}^{p - 1} \cdot \left( {\frac{\quad}{t}i_{1}} \right) \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqesn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}}} + {\sum\limits_{p = 1}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{{qi}_{2}^{q - 1} \cdot \left( {\frac{\quad}{t}i_{2}} \right) \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}}}\quad + {\sum\limits_{p = 1}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{r \cdot i_{3}^{r - 1} \cdot \left( {\frac{\quad}{t}i_{3}} \right) \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}}} + {\sum\limits_{p = 1}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\omega \cdot n \cdot \left( {{g_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}}}} & (78)\end{matrix}$

Dividing by angular velocity: $\begin{matrix}{\frac{v_{\varphi}}{\omega} = {{\frac{i_{\varphi}}{\omega} \cdot R_{\varphi}} + {\left\lbrack {\frac{1}{\omega} \cdot \left( {\frac{\quad}{t}i_{1}} \right)} \right\rbrack \cdot {{{\sum\limits_{p = 1}^{P}{p \cdot i_{1}^{p - 1} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}}} + {\left\lbrack {\frac{1}{\omega} \cdot \left( {\frac{\quad}{t}i_{2}} \right)} \right\rbrack \cdot {\sum\limits_{p = 1}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{{qi}_{2}^{q - 1} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}}}} + {\left\lbrack {\frac{1}{\omega} \cdot \left( {\frac{\quad}{t}i_{3}} \right)} \right\rbrack \cdot {\sum\limits_{p = 1}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{r \cdot i_{3}^{r - 1} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{\left( {{g_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {h_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}}}} + {\sum\limits_{p = 1}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{s = 0}^{S}{i_{4}^{s} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{\varphi \quad {pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{\varphi \quad {pqrsn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}}}}}}} & (90)\end{matrix}$

The state variables assigned to the notional rotor phase are notobservable via the motor terminals. Hence, the nominally constant rotorcurrent must be lumped in with those model parameters that areobservable, as disclosed above with respect to the model formulated forthe αβ-FoR. The following identities are defined: $\begin{matrix}{{{\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot g_{\varphi \quad {pqrsn}}}} = {G_{\varphi \quad {pqrn}}\quad {for}\quad {all}\quad p}},{q\quad {and}\quad r}} & (91) \\{{{\sum\limits_{s = 0}^{S}{I_{4}^{s} \cdot h_{\varphi \quad {pqrsn}}}} = {H_{\varphi \quad {pqrn}}\quad {for}\quad {all}\quad p}},{q\quad {and}\quad r}} & (92)\end{matrix}$

Using the identities provided by Equations (91) and (92), Equation (90)yields: $\begin{matrix}{\frac{v_{\varphi}}{\omega} = \quad {\frac{i_{\varphi}}{\omega} \cdot {{{R_{\varphi} + {\left\lbrack {\frac{1}{\omega} \cdot \left( {\frac{\quad}{t}i_{1}} \right)} \right\rbrack \cdot \quad {\sum\limits_{p = 1}^{P}{{p \cdot {{i_{1}^{p - 1} \cdot \quad {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{n = 1}^{N}{\quad \left( {{G_{\varphi \quad {pqrn}} \cdot \quad {\sin \left( {n \cdot \theta} \right)}} +} \right.}}}}}}}}}{\left. {{{H_{\varphi \quad {pqrn}} \cdot \left. {\cos \quad\left( \quad {n \cdot \theta} \right)} \right)}\quad \ldots} + {\left\lbrack {\frac{1}{\omega} \cdot} \right.\left( {\frac{\quad}{t}i_{2}} \right)}} \right\rbrack \cdot {\sum\limits_{p = 1}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{{qi}_{2}^{q - 1} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{n = 1}^{N}{\left( {{G_{\varphi \quad {pqrn}} \cdot \left( {n \cdot \theta} \right)} +_{\varphi \quad {pqrn}} \cdot} \right.\left. {\cos \left( {n \cdot \theta} \right)} \right)\quad \ldots}}}}}}}}}}}} + {\left\lbrack {\frac{1}{\omega} \cdot \left( {\frac{\quad}{t}i_{3}} \right)} \right\rbrack \cdot {\sum\limits_{p = 1}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{r \cdot i_{3}^{r - 1} \cdot {\sum\limits_{n = 1}^{N}{\left( {{G_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {H_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}}} + {\sum\limits_{p = 1}^{P}{i_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{3}^{r} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}}}}}} & (93)\end{matrix}$

Using the identities presented in Equations (91) and (92), Equation (89)is also re-written: $\begin{matrix}{T = {{\sum\limits_{p = 1}^{P}{\frac{I_{1}^{p + 1}}{p + 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{G_{1{p00n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{1{p00n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}} + {\sum\limits_{p = 1}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{\frac{I_{2}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{G_{2{pq0n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{2{pq0n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}} + {\sum\limits_{p = 1}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{G_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{3{pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}}}} + {\sum\limits_{s = 0}^{S}{\frac{I_{f}^{s + 1}}{s + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{4000{sn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{4000{sn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}} & (94)\end{matrix}$

and in the particular case of a switched reluctance machine:$\begin{matrix}\begin{matrix}\begin{matrix}{T = {{\sum\limits_{p = 0}^{P}{\frac{I_{1}^{p + 1}}{p + 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{g_{1{p00n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{1{p00n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}} +}} \\{{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}\quad {\frac{I_{2}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{g_{2{pq0n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{2{pq0n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}} +}\end{matrix} \\{{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {h_{3{pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}\quad}\end{matrix} & (95)\end{matrix}$

Not all model parameters from the voltage fit are present in the finalexpressions for torque as a result of the path selected in evaluatingthe integral.

The approach implemented by the solver 34 to calculate the requiredcurrents to achieve desired motor behavior, such as smooth torque withangle sensitivity minimized, is similar in the abc-FoR as that disclosedabove with regard to the αβ-FoR. The main difference is that theJacobian is a three by three matrix with the third row elements beinggiven by some constriction upon the values which the abc-FoR currentsmay take. For the purposes of the current disclosure, it will be assumedthat any solution chosen will in some way minimize the sum of squares ofthe individual phase currents.

Suppose torque is calculated via coenergy using the torque model withabc-FoR set out in Equation (88): $\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{T = {{\sum\limits_{p = 0}^{P}{\frac{I_{1}^{p + 1}}{p + 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{G_{1{p00n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{1{p00n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}} +}} \\{{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}\quad {\frac{I_{2}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{G_{2{pq0n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{2{pq0n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}} +}\end{matrix} \\{{{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{G_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{3{pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}}}} +}\quad}\end{matrix} \\{\sum\limits_{s = 0}^{S}\quad {\frac{I_{f}^{s + 1}}{s + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{4000{sn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {{h_{4000{sn}} \cdot \sin}\left( {n \cdot \theta} \right)}} \right)}}}}\end{matrix} & (88)\end{matrix}$

The necessary partial derivatives or entries to the Jacobian, withrespect to i₁,i₂ and i₃, can now be derived. $\begin{matrix}\begin{matrix}\begin{matrix}{{\frac{\partial\quad}{\partial i_{1}}T} = {{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{n = 1}^{N}\quad {{n \cdot \left( {{G_{1{p00n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{1{p00n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}} +}} \\{{\sum\limits_{p = 0}^{P}{p \cdot I_{1}^{p - 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {\frac{I_{2}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{G_{2{pq0n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{2{pq0n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}} +}\end{matrix} \\{\sum\limits_{p = 0}^{P}{p \cdot I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {{H_{3{pqrn}} \cdot \sin}\left( {n \cdot \theta} \right)}} \right)}}}}}}}}\end{matrix} & (96) \\\begin{matrix}{{\frac{\partial\quad}{\partial i_{2}}T} = {{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{n = 1}^{N}\quad {{n \cdot \left( {{G_{2{pq0n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{2{pq0n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}} +}} \\{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{q \cdot I_{2}^{q - 1} \cdot {\sum\limits_{r = 0}^{R}\quad {\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {{H_{3{pqrn}} \cdot \sin}\left( {n \cdot \theta} \right)}} \right)}}}}}}}}\end{matrix} & (97) \\{{{\frac{\partial\quad}{\partial i_{3}}T} = {\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{3}^{r} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {{H_{3{pqrn}} \cdot \sin}\left( {n \cdot \theta} \right)}} \right)}}}}}}}}}\quad} & (98) \\\begin{matrix}\begin{matrix}\begin{matrix}{\quad {{Hence}:}\quad} \\{J_{11} = {{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{G_{1{p00n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{1{p00n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}} +}}\end{matrix} \\{{\sum\limits_{p = 1}^{P}{p \cdot I_{1}^{p - 1} \cdot {\sum\limits_{q = 0}^{Q}{\frac{I_{2}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{G_{2{pq0n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{2{pq0n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}} +}\end{matrix} \\{{\sum\limits_{p = 1}^{P}{p \cdot I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {{H_{3{pqrn}} \cdot \sin}\left( {n \cdot \theta} \right)}} \right)}}}}}}}}\quad}\end{matrix} & (99) \\\begin{matrix}{J_{12} = {{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{n = 1}^{N}{{n \cdot \left( {{G_{2{pq0n}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {H_{2{pq0n}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}} +}} \\{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{q \cdot I_{2}^{q - 1} \cdot {\sum\limits_{r = 0}^{R}{\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {{H_{3{pqrn}} \cdot \sin}\left( {n \cdot \theta} \right)}} \right)}}}}}}}}\end{matrix} & (100) \\{{J_{13} = {\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {I_{3}^{r} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{G_{3{pqrsn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {{H_{3{pqrn}} \cdot \sin}\left( {n \cdot \theta} \right)}} \right)}}}}}}}}}\quad} & (101)\end{matrix}$

As stated previously in Equation (56), sensitivity with respect to angleis given by: $\begin{matrix}{\frac{\partial\quad}{\partial\theta}T} & (56)\end{matrix}$

From Equation (88): $\begin{matrix}{{\frac{\partial\quad}{\partial\theta}T} = {{\sum\limits_{p = 0}^{P}{\frac{I_{1}^{p + 1}}{p + 1} \cdot {\sum\limits_{n = 1}^{N}{{n^{2} \cdot \left( {{{- G_{1{p00n}}} \cdot {\sin \left( {n \cdot \theta} \right)}} - {H_{1{p00n}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}} +}} & \quad \\{{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}\quad {\frac{I_{2}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{{n^{2} \cdot \left( {{{- G_{2{pq0n}}} \cdot {\sin \left( {n \cdot \theta} \right)}} - {H_{2{pq0n}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}} +} & \quad \\{{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{{n^{2} \cdot \left( {{{- G_{3{pqrsn}}} \cdot {\sin \left( {n \cdot \theta} \right)}} - {H_{3{pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}}}} +} & \quad \\{\sum\limits_{s = 0}^{S}\quad {\frac{I_{f}^{s + 1}}{s + 1} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \left( {{g_{4000{sn}} \cdot {\sin \left( {n \cdot \theta} \right)}} - {h_{4000{sn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}} & {(102)\quad}\end{matrix}$

Hence, the elements of the second row of the Jacobian are given by thefollowing expressions: $\begin{matrix}{J_{21} = {\frac{\partial\quad}{\partial i_{1}}\frac{\partial\quad}{\partial\theta}T}} & (103) \\{J_{22} = {\frac{\partial\quad}{\partial i_{2}}\frac{\partial\quad}{\partial\theta}T}} & (104) \\{J_{23} = {\frac{\partial\quad}{\partial i_{3}}\frac{\partial\quad}{\partial\theta}T}} & (105)\end{matrix}$

By changing the order of the partial derivatives in Equations(103)-(105): $\begin{matrix}{J_{21} = {\frac{\partial\quad}{\partial\theta}J_{11}}} & (106) \\{J_{22} = {\frac{\partial\quad}{\partial\theta}J_{12}}} & (107) \\{J_{23} = {\frac{\partial\quad}{\partial\theta}J_{13}}} & (108) \\\begin{matrix}\begin{matrix}\begin{matrix}{{Explicitly}:} \\{J_{21} = {{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{n = 1}^{N}{{n^{2} \cdot \left( {{{- G_{1{p00n}}} \cdot {\sin \left( {n \cdot \theta} \right)}} - {H_{1{p00n}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}} +}}\end{matrix} \\{{\sum\limits_{p = 0}^{P}{p \cdot I_{1}^{p - 1} \cdot {\sum\limits_{q = 0}^{Q}\quad {\frac{I_{2}^{q + 1}}{q + 1} \cdot {\sum\limits_{n = 1}^{N}{{n^{2} \cdot \left( {{{- G_{2{pq0n}}} \cdot {\sin \left( {n \cdot \theta} \right)}} - {H_{2{pq0n}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}} +}\end{matrix} \\{\sum\limits_{p = 0}^{P}{p \cdot I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}\quad {\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n^{2} \cdot \left( {{{- G_{3{pqrsn}}} \cdot {\sin \left( {n \cdot \theta} \right)}} - {H_{3{pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}\end{matrix} & (109) \\\begin{matrix}{J_{22} = {{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{n = 1}^{N}{{n^{2} \cdot \left( {{{- G_{2{pq0n}}} \cdot {\sin \left( {n \cdot \theta} \right)}} - {H_{2{pq0n}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}\quad \ldots}}}}}} +}} \\{\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{q \cdot I_{2}^{q - 1} \cdot {\sum\limits_{r = 0}^{R}\quad {\frac{I_{3}^{r + 1}}{r + 1} \cdot {\sum\limits_{n = 1}^{N}{n^{2} \cdot \left( {{{- G_{3{pqrsn}}} \cdot {\sin \left( {n \cdot \theta} \right)}} - {H_{3{pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}\end{matrix} & (110) \\{J_{23} = {\sum\limits_{p = 0}^{P}{I_{1}^{p} \cdot {\sum\limits_{q = 0}^{Q}{I_{2}^{q} \cdot {\sum\limits_{r = 0}^{R}{I_{3}^{r} \cdot {\sum\limits_{n = 1}^{N}{n^{2} \cdot \left( {{{- G_{3{pqrsn}}} \cdot {\sin \left( {n \cdot \theta} \right)}} - {H_{3{pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}} & (111)\end{matrix}$

The criterion used in selecting current can be to either balance thefeed or minimize the sum of squares. In the case of the least squaressum, the following is minimized: $\begin{matrix}{I_{ss} = {\sum\limits_{k = 1}^{3}i_{k}^{2}}} & (112)\end{matrix}$

A minimum is achieved when: $\begin{matrix}{{\frac{\partial\quad}{\partial i_{k}}I_{ss}} = {{2 \cdot i_{k}} = c}} & (113)\end{matrix}$

The third row of the Jacobian is then given by expressions of the form:$\begin{matrix}{J_{3k} = {{\frac{\partial^{2}\quad}{\partial i_{k}^{2}}I_{ss}} = 2}} & (114)\end{matrix}$

Cogging can be included into the solver 34 in the same way as describedabove in conjunction with the αβ-FoR smooth torque solver.

Moreover, in a manner similar to that disclosed with the αβ-FoR smoothtorque solver, the solution may be calculated across the angle intervalsin a given interval simultaneously, rather than in a point by pointmanner. For a particular torque and sensitivity demand, the solution isto be calculated at various angles:

θ(k) for all k=1, . . . ,N

The current column vector is given by:

I=(i _(a)(θ(1)). . . i _(a)(θ(N))i _(b)(θ(1)). . . i _(b)(θ(N))i_(c)(θ(1)). . . i _(c)(θ(N)))^(T)  (115)

If the new

I(n+1)=I(n)+Δ(n)  (116)

The k'th torque vector is a row vector defined by:

 φ_(Tk)=(0 . . . 0 T(θ(k),i _(a)(k),i _(b)(k),i _(c)(k))0 . . . 0)

Similiarly the kth sensitivity vector is defined by:

φ_(Sk)=(0 . . . 0 S(θ(k),i _(a)(k),i _(b)(k),i _(c)(k))0 . . . 0)

Finally, the minimum sum of squares current is given by:

φ_(Ik)=(0 . . . 0 I_(ss)(i _(a)(k),i _(b)(k),i _(c)(k))0 . . . 0)

These vectors can be stacked to form diagonal matrices:$A = {{\begin{pmatrix}\varphi_{T1} \\\cdots \\\varphi_{TN}\end{pmatrix}\quad B} = {{\begin{pmatrix}\varphi_{S1} \\\cdots \\\varphi_{SN}\end{pmatrix}\quad C} = \begin{pmatrix}\varphi_{I1} \\\cdots \\\varphi_{IN}\end{pmatrix}}}$

Thus, appropriate partial derivatives with respect to the currents aretaken and the resultant matrices aggregated to form a 3N by 3N matrix:$\begin{matrix}{\Phi = \begin{pmatrix}{\frac{\partial\quad}{\partial i_{a}}A} & {\frac{\partial\quad}{\partial i_{b}}A} & {\frac{\partial\quad}{\partial i_{c}}A} \\{\frac{\partial\quad}{\partial i_{a}}A} & {\frac{\partial\quad}{\partial i_{b}}B} & {\frac{\partial\quad}{\partial i_{c}}C} \\{\frac{\partial\quad}{\partial i_{a}}A} & {\frac{\partial\quad}{\partial i_{b}}B} & {\frac{\partial\quad}{\partial i_{c}}C}\end{pmatrix}} & (117)\end{matrix}$

The desired torque, sensitivity and rate of change of the sum of squaresat a particular angle θ(k) are given by:

T(θ(k)) S(θ(k)) I(θ(k))

The demand vector D of these values over the angle range is given by:$\begin{matrix}{D = \begin{pmatrix}{T_{d}\left( {\theta (1)} \right)} \\{T_{d}\left( {\theta (2)} \right)} \\\cdots \\{T_{d}\left( {\theta (N)} \right)} \\{S_{d}\left( {\theta (1)} \right)} \\{S_{d}\left( {\theta (2)} \right)} \\\cdots \\{S_{d}\left( {\theta (N)} \right)} \\{I\left( {\theta (1)} \right)} \\\cdots \\{I\left( {\theta (N)} \right)}\end{pmatrix}} & (118)\end{matrix}$

and the actual values of torque and sensitivity resultant from anycurrent combination (i_(a),i_(b),i_(c)) which constitute the iteratedsolution are given by the column vector: $\begin{matrix}{A = \begin{pmatrix}{T\quad \left( {{\theta (1)},{i_{a}(1)},{i_{b}(1)},{i_{c}(1)}} \right)} \\{T\left( {{\theta (2)},{i_{a}(2)},{i_{b}(2)},{i_{c}(2)}} \right)} \\\cdots \\{T\left( {{\theta (N)},{i_{a}(N)},{i_{b}(N)},{i_{c}(N)}} \right)} \\{S\left( {{\theta (1)},{i_{a}(1)},{i_{b}(1)},{i_{c}(1)}} \right)} \\\cdots \\{S\left( {{\theta (N)},{i_{a}(N)},{i_{b}(N)},{i_{c}(N)}} \right)}\end{pmatrix}} & (119)\end{matrix}$

Using this notation and from the development initially presented:

Δ(n)=Φ⁻¹·(D−A)  (120)

with:

 I(n+1)=I(n)+Δ(n)

In one embodiment of the invention, a PM motor was used with a modelfitted via the terminal variables as described herein. From this, smoothtorque feeds were calculated for a variety of loads. FIG. 4 illustratesthe calculated current profiles for various torques, showing three phasecurrents generated for a smooth torque solution. FIG. 5 shows a plot fora typical 12-10 PM motor, for which excluding noise and peaks (the plotillustrates raw data) ripple approaches 2% of the mean or 0.8% of themaximum rated torque for that motor (2.5 Nm).

It has been assumed that the individual components of the electricalmodel and correspondingly the torque model comprise products ofpolynomials and trigonometric functions. In alternative embodiments, thepolynomials are replaced with true orthogonal functions. Theseorthogonal functions are built up from polynomials in a recursivemanner. In particular polynomials of the form:

1,x,x²,x³,x⁴,x⁵

are replaced with expressions of the form:$\frac{1}{\sqrt{2}},{\sqrt{\frac{3}{2}} \cdot x},{\sqrt{\frac{5}{8}} \cdot \left( {{3x^{2}} - 1} \right)},{\sqrt{\frac{7}{8}} \cdot \left( {{5 \cdot x^{3}} - {3 \cdot x}} \right)},{\frac{3}{8 \cdot \sqrt{2}} \cdot \left( {{35 \cdot x^{4}} + 3 - {30 \cdot x^{2}}} \right)},{\sqrt{\frac{43659}{128}} \cdot \left( {x^{5} - {\frac{70}{63} \cdot x^{3}} + {\frac{15}{63} \cdot x}} \right)}$

There are a number of sound theoretical and practical reasons why modelsusing true orthogonal functions are to be preferred. Typically, moreaccurate models that have fewer terms can be derived. Such a statementis true when the order of the current terms grows beyond 2. Thetheoretical reason for this is well understood by those with anunderstanding of such mathematical structures. Succinctly, modelcomponents that are orthogonal to one another do not interact in adetrimental manner. Unnecessary model complexity is avoided and theresult is that the torque estimate, achieved via the previouslydescribed transforms, are improved.

The process necessary to derive the mathematical expressions necessaryis essentially the same as those described previously. For the purposesof brevity, the critical mathematical expressions and notation arepresented without unnecessary repetition of the associated derivationsfor the balanced feed case.

Let the r'th order orthogonal function of variable x be:

g_(r)(x)

The expression for flux is then:${\lambda_{\varphi}\left( {i_{\alpha},i_{\beta},\theta} \right)} = {\sum\limits_{p = 0}^{P}\quad {{g_{p}\left( i_{\alpha} \right)} \cdot {\sum\limits_{q = 0}^{Q}\quad {{g_{q}\left( i_{\beta} \right)} \cdot {\sum\limits_{r = 0}^{R}\quad {{g_{r}\left( i_{f} \right)} \cdot {\sum\limits_{n = 0}^{N}\quad \left( {{a_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {b_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)}}}}}}}$

The derivative of flux is given by: $\begin{matrix}{{\frac{\quad}{t}\lambda_{\varphi}} = {{\sum\limits_{p = 0}^{P}\quad {{f_{p}\left( i_{\alpha} \right)} \cdot \left( {\frac{\quad}{t}i_{\alpha}} \right) \cdot {\sum\limits_{q = 0}^{Q}\quad {{g_{q}\left( i_{\beta} \right)}{\sum\limits_{r = 0}^{R}\quad {{g_{r}\left( i_{f} \right)} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{a_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {b_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}} +}} & \quad \\{{\sum\limits_{p = 0}^{P}\quad {{g_{p}\left( i_{\alpha} \right)} \cdot {\sum\limits_{q = 0}^{Q}\quad {{f_{q}\left( i_{\beta} \right)} \cdot \left( {\frac{\quad}{t}i_{\beta}} \right) \cdot {\sum\limits_{r = 0}^{R}\quad {{g_{p}\left( i_{f} \right)} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{a_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {b_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}} +} & \quad \\{{\sum\limits_{p = 0}^{P}\quad {{g_{p}\left( i_{\alpha} \right)} \cdot {\sum\limits_{q = 0}^{Q}\quad {{g_{q}\left( i_{\beta} \right)} \cdot {\sum\limits_{r = 0}^{R}\quad {{f_{r}\left( i_{\beta} \right)} \cdot \left( {\frac{\quad}{t}i_{f}} \right) \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{a_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {b_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}}} +} & \quad \\{{\sum\limits_{p = 0}^{P}\quad {{g_{p}\left( i_{\alpha} \right)} \cdot {\sum\limits_{q = 0}^{Q}\quad {{g_{q}\left( i_{\beta} \right)} \cdot {\sum\limits_{r = 0}^{R}\quad {{g_{r}\left( i_{f} \right)} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \omega \cdot \left( {{a_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {b_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}}}\quad} & \quad \\{{{where}:\quad {\frac{\quad}{x}{g_{p}(x)}}} = {f_{p}(x)}} & \quad\end{matrix}$

Then, the electrical equation is written: $\begin{matrix}\begin{matrix}\begin{matrix}{\frac{v_{\varphi}}{\omega} = {\frac{R_{\varphi}}{\omega} + {\frac{i_{\alpha}}{\omega} \cdot R_{\varphi \quad \alpha}} + {\frac{i_{\beta}}{\omega} \cdot R_{\varphi \quad \beta}} + {{\frac{i_{\alpha} \cdot i_{\beta}}{\omega} \cdot R_{{\varphi \quad \alpha \quad \beta}\quad}}\quad \ldots} +}} \\{{\left\lbrack {\frac{1}{\omega} \cdot \left( {\frac{\quad}{t}i_{\alpha}} \right)} \right\rbrack \cdot {\sum\limits_{p = 0}^{P}\quad {{f_{p}\left( i_{\alpha} \right)} \cdot {\sum\limits_{q = 0}^{Q}\quad {{g_{q}\left( i_{\beta} \right)} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{A_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {B_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}} +}\end{matrix} \\{{\left\lbrack {\frac{1}{\omega} \cdot \left( {\frac{\quad}{t}i_{\beta}} \right)} \right\rbrack \cdot {\sum\limits_{p = 0}^{P}\quad {{g_{p}\left( i_{\alpha} \right)} \cdot {\sum\limits_{q = 0}^{Q}\quad {{f_{q}\left( i_{\beta} \right)} \cdot {\sum\limits_{n = 0}^{N}\quad {\left( {{A_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}} + {B_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}}} \right)\quad \ldots}}}}}}} +}\end{matrix} \\{\sum\limits_{p = 0}^{P}\quad {{g_{p}\left( i_{\alpha} \right)} \cdot {\sum\limits_{q = 0}^{Q}\quad {{g_{q}\left( i_{\beta} \right)} \cdot {\sum\limits_{n = 1}^{N}{n \cdot \quad \left( {{A_{\varphi \quad {pqrn}} \cdot {\cos \left( {n \cdot \theta} \right)}} - {B_{\varphi \quad {pqrn}} \cdot {\sin \left( {n \cdot \theta} \right)}}} \right)}}}}}}\end{matrix}$

Coenergy is derived in a similar manner as previously resulting in:$\begin{matrix}{\omega_{c} = {{\sum\limits_{p = 0}^{P}\quad {\left( {{h_{p}\left( I_{\alpha} \right)} - {h_{p}(0)}} \right) \cdot {\sum\limits_{q = 0}^{Q}\quad {{g_{q}\left( I_{\beta} \right)} \cdot {\sum\limits_{n = 0}^{N}{\begin{bmatrix}{{{\left( {A_{a\quad {pqrn}} - {\frac{1}{2} \cdot A_{b\quad {pqrn}}} - {\frac{1}{2} \cdot A_{c\quad {pqrn}}}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{{\left( {B_{a\quad {pqrn}} - {\frac{1}{2} \cdot B_{b\quad {pqrn}}} - {\frac{1}{2} \cdot B_{c\quad {pqrn}}}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad}\end{bmatrix}\quad \ldots}}}}}} +}} \\{{\sum\limits_{p = 0}^{P}{{g_{p}(0)} \cdot {\sum\limits_{q = 0}^{Q}\quad {\left( {{h_{q}\left( I_{\beta} \right)} - {h_{q}(0)}} \right) \cdot {\sum\limits_{n = 0}^{N}\begin{bmatrix}{{{\left( {{\frac{- \sqrt{3}}{2} \cdot A_{b\quad {pqrn}}} + {\frac{\sqrt{3}}{2} \cdot A_{c\quad {pqrn}}}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{{\left( {{\frac{- \sqrt{3}}{2} \cdot A_{b\quad {pqrn}}} + {\frac{\sqrt{3}}{2} \cdot A_{c\quad {pqrn}}}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad}\end{bmatrix}}}}}}\quad}\end{matrix}$

where:

h _(p)(x)=∫f _(p)(x)dx

Then, torque is given by: $\begin{matrix}{T = {{\sum\limits_{p = 0}^{P}\quad {\left( {{h_{p}\left( I_{\alpha} \right)} - {h_{p}(0)}} \right) \cdot {\sum\limits_{q = 0}^{Q}\quad {{g_{q}\left( I_{\beta} \right)} \cdot {\sum\limits_{n = 0}^{N}{{n \cdot \begin{bmatrix}{{{\left( {A_{a\quad {pqrn}} - {\frac{1}{2} \cdot A_{b\quad {pqrn}}} - {\frac{1}{2} \cdot A_{c\quad {pqrn}}}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{{{- \left( {B_{a\quad {pqrn}} - {\frac{1}{2} \cdot B_{b\quad {pqrn}}} - {\frac{1}{2} \cdot B_{c\quad {pqrn}}}} \right)} \cdot {\sin \left( {n \cdot \theta} \right)}}\quad}\end{bmatrix}}\quad \ldots}}}}}} +}} \\{{\sum\limits_{p = 0}^{P}{{g_{p}(0)} \cdot {\sum\limits_{q = 0}^{Q}\quad {\left( {{h_{q}\left( I_{\beta} \right)} - {h_{q}(0)}} \right) \cdot {\sum\limits_{n = 0}^{N}{n \cdot \begin{bmatrix}{{{\left( {{\frac{- \sqrt{3}}{2} \cdot A_{b\quad {pqrn}}} + {\frac{\sqrt{3}}{2} \cdot A_{c\quad {pqrn}}}} \right) \cdot {\cos \left( {n \cdot \theta} \right)}}\quad \ldots} +} \\{{\left( {{\frac{- \sqrt{3}}{2} \cdot A_{b\quad {pqrn}}} + {\frac{\sqrt{3}}{2} \cdot A_{c\quad {pqrn}}}} \right) \cdot {\sin \left( {n \cdot \theta} \right)}}\quad}\end{bmatrix}}}}}}}\quad}\end{matrix}$

As previously, a solver can now be defined which will calculate thenecessary current values to achieve the desired solution.

One property and advantage associated with the use of model componentswhich are truly orthonormal is that with just the most basic modelpresent it is possible to calculate the parameters of other additionalorthonormal model components in isolation from one another. That is, themodel can be refined by the addition of a new orthonormal expression.The parameters for the model components estimated and the new modelcomponent can be either kept or discarded dependant upon the magnitudeof the parameter associated with it.

In this manner, many different model components can be sieved in anumerically efficient and elegant manner to determine whether theyshould be included in the model or not. More particularly, the modelfitting process can be fully automated. Starting from a very basic model(the presence of resistance terms, non-angle varying inductance) it ispossible to automate the selection of model components. Starting from abasic model as described above, the electrical model is extended byselecting one or more additional “candidate” basis functions. Refitingthe model results in a new set of model parameters.

Those parts of the model, or functions, with significant coefficients orparameters are kept while other parts are rejected. The test forsignificant a model component can be as simple as testing whether theabsolute value of the coefficient is greater than some predefinedpercentage of the absolute value of the current biggest parameter. Sucha sieving process allows for the automatic building up of a model. It isthe use of orthonormal basis functions which allows for this activity,due to their minimal interaction. If the “standard” polynomials are usedthen the sieving process becomes confused.

The adaptive control schemes disclosed herein have several applications.For example, in accordance with certain embodiments of the invention,the control scheme is embedded into a speed control loop for use in aspeed servo application. The particular embodiments disclosed above areillustrative only, as the invention may be modified and practiced indifferent but equivalent manners apparent to those skilled in the arthaving the benefit of the teachings herein. For example, the electricalmodel which uses a product of polynomials and trigonometric functionscan be written as a product of polynomials and complex exponentials:$\lambda_{\varphi} = {\sum\limits_{p = 0}^{P}{i_{\alpha}^{p} \cdot {\sum\limits_{q = 0}^{Q}{i_{\beta}^{q} \cdot {\sum\limits_{r = 0}^{R}{i_{f}^{r} \cdot {\sum\limits_{n = {- N}}^{N}{e_{\varphi \quad {pqrn}} \cdot {\exp \left( { \cdot n \cdot \theta} \right)}}}}}}}}}$

Furthermore, no limitations are intended to the details of constructionor design herein shown, other than as described in the claims below. Itis therefore evident that the particular embodiments disclosed above maybe altered or modified and all such variations are considered within thescope and spirit of the invention. Accordingly, the protection soughtherein is as set forth in the claims below.

What is claimed is:
 1. A method for controlling a rotatingelectromagnetic machine, the machine including a stator and a rotor thatrotates relative to the stator, the stator including a plurality ofphase windings, the method comprising: receiving feedback regarding therotor position relative to the stator; receiving feedback regardingenergization of the phase windings; developing a first mathematicalmodel based on the rotor position and energization feedback to describeelectrical behavior of the machine; developing a second mathematicalmodel via a mathematical transform of the first mathematical model todescribe torque characteristics of the machine; receiving a torquedemand signal; and calculating a phase energization current value viathe second mathematical model and the torque demand signal.
 2. Themethod of claim 1, further comprising energizing the phase windings withthe calculated phase energization current value.
 3. The method of claim2, wherein the phase windings are energized with a balanced feed.
 4. Themethod of claim 1, wherein the first mathematical model is non-linear.5. The method of claim 3, wherein the electrical behavior described bythe first mathematical model includes the relationship between voltage,current and rotor position over the predetermined operating range. 6.The method of claim 1, wherein the first mathematical model describeselectrical behavior of the machine over a predetermined operating range.7. The method of claim 1, wherein the first and second mathematicalmodels include components that comprise products of polynomials andtrigonometric functions.
 8. The method of claim 1, wherein the first andsecond mathematical models include components that comprise orthogonalfunctions.
 9. The method of claim 1, wherein the first mathematicalmodel includes parameters estimated recursively.
 10. The method of claim8, wherein collected data older than a predetermined age is not used inparameter estimation.
 11. The method of claim 1, further comprisinggenerating a lookup table correlating torque demand values with thephase energization current values.
 12. The method of claim 1, whereincalculating the phase energization current value includes calculatingthe phase energization current value in accordance with a desiredmachine behavior.
 13. The method of claim 12, wherein the desiredmachine behavior includes minimizing torque ripple.
 14. The method ofclaim 13, wherein the phase energization current is further calculatedto reduce sensitivity to rotor position measurement errors.
 15. Themethod of claim 1 further comprising updating the second mathematicalmodel aft predetermined times.
 16. The method of claim 15, wherein thepredetermined times occur when the machine is not operating.
 17. Themethod of claim 1, wherein developing the second mathematical modelincludes modeling the non-load dependent cogging torque.
 18. The methodof claim 17, wherein modeling the non-load dependent cogging torquecomprises: spinning the rotor unloaded at a predetermined angularvelocity; measuring the phase windings voltage and current: determiningthe rotor positions associated with the voltage and currentmeasurements; developing a first mathematical model based on themeasured voltage and rotor position to describe electrical behavior ofthe machine; developing a second mathematical model via a mathematicaltransform of the first mathematical model to describe torquecharacteristics of the machine; energizing the windings such that therotor holds a predetermined position; and calculating the cogging torquefor the predetermined position via the second mathematical model.
 19. Asystem for controlling a rotating electromagnetic machine including astator having a plurality of phase windings, a rotor that rotatesrelative to the stator, and a drive connected to the phase windings forenergizing the windings, the control system comprising: an estimatorconnectable so the machine for receiving signals representing the phasewinding voltage and rotor position; the estimator outputting parameterestimations for an electrical model of the machine based on the receivedvoltage and rotor position; a torque model receiving the parameterestimations from the estimator, the torque model outputting estimates oftorque for associated rotor position-phase current combinations; and acontroller having input terminals for receiving a torque demand signaland the rotor position signal, the controller adapted to output acontrol signal to the drive in response to the torque demand and rotorposition signals and the torque model.
 20. The system of claim 19,wherein the estimator outputs the model parameters at predeterminedtimes.
 21. The system of claim 20, wherein the predetermined timescomprise times when the rotating machine is not operational.
 22. Thesystem of claim 19, further comprising a solver coupled to the torquemodel and the controller, the solver calculating phase current profilesto energize the machine to achieve a predetermined machine behavior. 23.The system of claim 22, wherein the predetermined machine behaviorincludes minimizing torque ripple.
 24. The system of claim 23, whereinthe phase energization current is further calculated to reducesensitivity to rotor position measurement errors.
 25. The system ofclaim 22, wherein the current profiles calculated by the solver arestored in a lookup table accessible by the controller.
 26. The system ofclaim 22, wherein the solver updates the current profiles atpredetermined times.
 27. A rotating electromagnetic machine system,comprising: a stator; a plurality of phase windings situated in thestator, a rotor situated so as to rotate relative to the stator, a rotorposition sensor outputting a signal representing the rotor positionrelative to the stator, a drive connected to the phase windings forenergizing the windings; an estimator connected to the phase windingsand the rotor position sensor to receive signals representing the phasewinding voltage and rotor position, the estimator outputting parameterestimations for an electrical model of the machine based on the receivedvoltage and rotor position; a torque model receiving the parameterestimations from the estimator, the torque model outputting estimates oftorque for associated rotor position-phase current combinations; and acontroller having input terminals for receiving a torque demand signaland the rotor position signal, the controller adapted to output acontrol signal to the drive in response to the torque demand and rotorposition signals and the torque model.
 28. The system of claim 27,further comprising a solver coupled to the torque model and thecontroller the solver calculating phase current profiles to energize thephase windings to achieve a predetermined machine behavior.
 29. Thesystem of claim 28, wherein the current profiles calculated by thesolver me stored in a lookup table accessible by the controller.
 30. Thesystem of claim 27, wherein the driver energizes the phase winding: witha balanced feed.
 31. A method of determining a non-load dependentcogging torque in a permanent magnet motor having a stator, a pluralityof stator windings having terminals connectable to a source of power forenergizing the windings, a rotor situated to rotate relative to thestator, the method comprising: spinning the rotor unloaded at apredetermined angular velocity; measuring voltage and current at themotor terminals; determining the rotor positions associated with thevoltage and currant measurements; developing a first mathematical modelbased on the measured voltage and rotor position to describe electricalbehavior of the machine; developing a second mathematical model via amathematical transform of the first mathematical model to describetorque characteristics of the machine; energizing the windings such thatthe rotor holds a predetermined position; and calculating the coggingtorque for the predetermined position via the second mathematical model.32. The method of claim 31, further comprising calculating the coggingtorque for a plurality of predetermined positions about a completerevolution of the rotor.
 33. The method of claim 31, wherein spinningthe rotor unloaded comprises spinning the rotor unenergized.
 34. Themethod of claim 31, wherein the rotor is spun at a plurality ofpredetermined angular velocities.
 35. A system for controlling arotating electromagnetic machine including a stator having a pluralityof phase windings, a rotor that rotates relative to the stator, and adrive connected to the phase windings for energizing the windings thecontrol system comprising: an estimator connectable to the machine forreceiving signals representing the phase winding voltage and rotorposition; the estimator outputting parameter estimations for anelectrical model of the machine based on the received voltage and rotorposition at predetermined times, the predetermined times including timeswhen the rotating machine is not operational; a torque model receivingthe parameter estimations from the estimator, the torque modeloutputting estimates of torque the associated rotor position-phasecurrent combinations; and a controller having input terminals forreceiving a torque demand signal and the rotor position signal, thecontroller adapted to output a control signal to the drive in responseto the torque demand and rotor position signals and the torque model.